Well, we don't know whether Caesar would indeed
recognise Caesarion as his own son or treat him as such (also, I think that the whole "Caesarion is the son of Caesar" trope really comes from Antony's declaration in Alexandria during the Donations ceremony in 33 BC. and that apart from that, there isn't any substantial evidence to support this claim - although the date of birth seems to fit and I may be wrong of course).
For one thing, Caesar will not be as dependent on Egypt as Antony was: he has already prepared the army and the funds for the campaign (stored in Apollonia); he can dedicate all his energies in the Parthian campaign and he has the resources of the whole Roman world at his disposal. Therefore, he wouldn't have to make so many concessions to Cleopatra. The question that still arises though is how much of the OTL Donations of Alexandria was the result of Cleopatra's influence on Antony and Antony's dependence on Egypt and how much was the result of Antony's own political designs and calculations.
If we assume that the Donations were really a "lover's gift" from Antony to Cleopatra and/or a "gift" aimed at ensuring Egypt's continued support, then we can safely assume that Caesar wouldn't be so generous. He didn't seem to be as defferential as Antony; in fact, it could be argued that he placed quite some checks on her power as queen - he allowed Arsinoe to remain in Ephesus, which I think was a measure aimed at keeping Egypt on a short leash by using her as a threat (at least unofficially) and the fact that Ptolemy XIV wasn't killed until after Caesar's assassination seems to me -at least- as an indication of Caesar not being really supportive of Cleopatra taking this step and becoming the sole ruler of Egypt. He also was in a much more independent position than Antony, as mentioned above. As such, I think that he would allow Cleopatra to rule alongside her brother; perhaps he would insist on making Caesarion the heir to the throne and even give him Cyprus to rule until his time to become king in Alexandria came.
But there is the possibility of Caesar having a greater role for Egypt in mind, but most probably in a different way than Antony: Caesar would clearly be on the offensive throughout the campaign while Antony was forced to remain on the defensive for almost four years; until Caesar, the only defeat Rome had sustained at the hands of the Parthians was at Carrhae and this was outside Roman territory - when Roman defenses were tested, they proved to be effective (Cassius repelled the Parthian invasion of Syria shortly after Carrhae, Cicero managed to do the same in Cilicia in 51 BC.), while Antony witnessed a near meltdown of Roman defenses in Asia (true, the area was very lightly defended, but still), with the Parthians capturing most of Syria and southern Asia Minor; and last, but not least, as it has been already mentioned, the personal relationship of each man with Cleopatra was different.
What all this means is that Caesar might envision a future where Egypt is one of the centrepieces of Rome's eastern policy (perhaps even THE centrepiece), but not the extent that Antony did. Antony must have draw the conclusion that the system that Pompey had set up in 63-62 BC. was ineffective and he might have gravitated towards a revamped system, where the myriad of clients kingdoms, principalities and cities would be replaced by a number of strong client states, particularly near the border with Parthia. Egypt met the requirements, as it was the wealthiest and most powerful of the client kingdoms, while it also had a veneer of legitimacy to rule over various regions in the East as Cleopatra was a descendant of the Seleucid dynasty as well; combined with Antony's personal history, it was the ideal option.
Caesar on the other hand wouldn't probably go down that direction (given the relative lack of information about his intentions regading Parthia, I will be mostly conjecturing - sorry for that); and what he would do wold depend on the outcome of the campaign:
- So, let's assume that the campaign commences in the spring of 41 BC. after having dealt with Dacia, Caesar invades Parthia; let's also move the dynastic troues of Orodes IV et co. to around that period, in order to maximise Caesar's chances of winning. It would take at least two years for the campaign to be over; in the end, Media, Mesopotamia, Babylonia and Susiana are captured by the Romans, with the Parthians being forced to retreat east of the Caspian Gates. I think that Caesar would certainly make annexations (Mesopotamia and Babylonia, alongside Susiana to protect the Romans in the south). Characene would become a client kingdom; Persis and Elymais would also probably go down the same route, as would Media Atropatene, which might be rewarded with Media proper. Therefore, by 40-39 BC., we have a weakened Parthian realm and the Roman sphere of influence encompassing much of Iran.
In such a senario, Caesar would probably invest much energy towards building up the new frontline states; Persis and Media would be two states which, combined and with some Roman backing, could become an effective buffer against a Parthian realm reduced to Parthia and the nearby provinces. As the Parthian threat has been pushed way too east, I don't think that Caesar would feel the need to empower any state west of the Euphrates. Egypt would most likely remain independent and retain Cyprus - but that's it. I don't think Caesar would go on transferring control of Roman provinces, such as Cyrenaica, to Egypt, neither would he force neigbouring client kings and dynasts to submit to Egyptian demands, as this could potentially destablise the East. He wouldn't have to, as Roman rule who seem to be fairly secure and the existing system to be functional
- On the other hand, given that Caesar would be almost 60 years old by that time, it is possible that his performance wouldn't be stellar: the Romans may manage to secure most of the lands west of the Zagros mountains, but the Parthians could still remain a potent force, ready to strike back. The result of this campaign would be more like a (very) favourable truce for thr Romans rather than a decisive victory. Therefore, Caesar would most probably try to create the conditions for Rome's eventual victory (he might even plan of leading a second offensive to break the Parthians). As such, he would want to bind the client kingdoms more closely to Rome and organise them better, in order to be able to muster an even larger force in the future war, a prerequisite to achieve a decisive victory, as well as secure the needed resources. Furthermore, he would have to ensure that the new order of things in the east was at least theoretically legitimate, in order to strengthen its foundations: the client system established by Pompey hadn't been stellar, as between 63 and 31 BC. , it became clear that it was often unstable (as was seen in Armenia, Cappadocia and Judea); an arrangement more well received by the locals might help in this. Finally, he would find it helpful if he were to be able to undermine the rule of the Parthian king and also somehow shoot down the Arsakid claims of imperial rights over the Near East and deprive the latter of support. Under these circumstances, I think that he might conceive something mirroring Antony's plans, with Egypt becoming Rome's junior partner in the east: Egypt could become the stabilising factor, thanks to its wealth and overall power; Cleopatra (and Caesarion) were descendants of the Seleucid dynasty as well and could therefore be considered to be legitimate rulers of the East, which would also help against the ideological foundations of Parthia's expansionism at the time. Last, but not least, Caesar might have not wanted to stretch Rome's administrative and military resources too thin; at the same time, the possibility of a governor ruling over a wealthy province fairly away from Rome (and therefore, more difficult to be controlled), who would also happen to be at the head of a substantial army, given the defense needs of the new provinces east of the Euphrates wouldn't be something Caesar would take lightly, especially since he had almost been in the same position when he crossed the Rubicon and he would probably grow more suspicious with age, as the prestige of his victories would start waning -or so he would think, and therefore would consider his position weaker.
All of these reasons could push Caesar towards a policy of less direct rule in Asia. He could create a client kingdom out of Mesopotamia, Babylonia and Susiana which could be given to Ptolemy XIV, if Cleopatra managed to convince Caesar to leave her as sole ruler of Egypt. I don't think that he would give it to Caesarion (Cleopatra would most likely oppose this, as she would want Caesarion to inherit the Ptolemaic throne and Caesar might wish to avoid some gossip back to Rome about the favour he shows to his alleged son). This is of course the more sensible route; Caesar could as well go full "grand plan" mode and put Rome' presence in the East on entirely new foundations, by creating a kingdom out of Mesopotamia, Babylonia, Susiana, Syria and perhaps Cilicia, that would become a sort of successor to the Seleucid monarchy, which Cleopatra might manage to convince him to entrust to Caesarion. Although with hindsight this might seem to be a good move (it creates another strong ally for Rome in the region, strong enough to be able to stand up against the Parthians -at least theoretically, Rome remains the de facto master of the East, the Romans simply revert to a policy more akin to what they did before 133 BC., Rome's administrative machine isn't too taxed and its military resources aren't spread), many in Rome would perceive this as treason, if not proof of Caesar's mind going soft, both of which could lead to a conspiracy to remove Caesar from his position of power.
Now, about Octavian, it really depends on what will happen while he is in the East with his uncle, the way Caesar dies and the aftermath of his death.
I think Octavian would gradually become a sort of unofficial political advisor to Caesar (primarily because politics was largely his element). In that capacity, as well as that of the designated (? - I don't know whether Caesar would inform him of the contens of his will) successor, I could see him coming to conflict with Cleopatra, as he would try to preserve his uncle's political position (and by extension his) back in Rome and therefore, he would oppose any move that might be miscontrued as "scandalous favouritism" towards the Egyptian queen and also, to protect his own position against potential claims of Cleopatra that Caesarion was Caesar's son.
If Caesar dies from natural causes (not impossible given his age, since he would be campaigning for nearly 3-4 years and not in the most hospitable places), then I think that Octavian would be in a stronger position, with him being more widely known; perhaps Caesar might have presented him as his heir before or shotly after the campaign. Still though, I think that at least in the beginning, he would have to share power with some other leading Caesarians, until he manages to become the undisputed leader of the faction. In this case, I think that he might feel confident enough, especially if succeds in positioning hiself as leader of Rome, to allow Caesarion to rule in Egypt after her mother's death, if nothing troublesome hadn't happened in the meantime. If, on the other hand, Caesar still ends up assassinated after the campaign and there is a new round of civil wars, he would most likely need the military support of already established military leaders of the Caesarian faction, which could lead to something like the OTL Triumvirate. In that case, I think it's very possible for Octavian to try to use the slightest excuse available to push for an intervention in Egypt - although he might not get his way on this one, since Cleopatra would most likely play it safe; however, this would be only temporary, as Octavian would want to eliminate any potential threat to his position.
(just my views)
