Following Alexander's death, the Macedonian Empire entered a Golden Age - the Pax Macedonica according to the Romans (a people more or less irrelevant in this world) - which was an age of lasting peace and prosperity throughout the realm. It was also a time of massive cultural exchanges as much of the known world was Hellenized in various degrees. Expansion of the Empire - the Oikoumene - ceased by the time Alexander's grandson Alexander V took the throne, but the spread of Hellenistic culture did not cease. Greek became the lingua franca and could be heard by traders and diplomats as far apart as Britain and the East Indies by the 1st Century AD. Greek architecture was constructed by Gaul and Parthian alike. Greek philosophical ideas influenced many from Egypt to Babylon to Persia and beyond. Interestingly, the ideas of foreign cultures also affected the Greeks in many ways thanks to Alexander's policy of assimilating Greek soldiers and elites with non-greek elites.
Religion in the Empire was eventually fixed into a modified Hellenic religion. The Cult of Alexander formed the basis of the civil religion which supplemented the old gods. Indeed many aspects of the Alexander cult were inspired by the old god-kings of Egypt and Mesopotamia. Also, initially Stoicism and then Neoplatonism a few centuries later became the dominant intellectual foundation of the religion, at least to the elites and intellectuals. Non-hellenic ideas came to affect the popular religion as well. Reincarnation from the Indians became popular, but also especially Iranian and Semitic ideas of the Apocalypse, Last Judgement and coming of the Saviour at the End of Days. (Of course, the Saviour is expected to be the reincarnation of Alexander who would come again in glory to conquer the entire world definitively, judge the nations, institute cultic reform and bring the mythic Golden Age back to earth.)
Peace and prosperity were good - but it could not last forever. Eventually, the Empire grew weak, divided. Her elites became depraved and lacked the unity that drove their ancestors to conquer the world, and then overcome by plague and new ambitious rivals, the Empire eventually fell in the second century AD.
In the coming centuries Carthage began to truly blossom as Macedonian power waned. Her captains succeeded in punicizing Hispania and chastising the upstart Romans effectively. But then, her explorers and merchants accidentally discovered the New World. This accident brought new opportunities to the Carthaginians, no stranger to opportunities indeed. Trade contacts with the natives began, slowly at first but then much more. Settlements were established at lucrative sites. Carthage had no interest in conquering the peoples of the New World, which gave the natives much needed time and space to recover from the Old World diseases that brought disaster to their peoples.
What befell the peoples of the New World was nothing short of a tragedy. With time however, the New World began to recover and reach their historic potential in the face of Carthaginian trade dominance and visits from other inquisitive sailors and explorers like the Norse and Celts.
The Maya experienced their Classic Era, differently perhaps, but still a time of cultural flourishing. The peoples of the Mississippi region and Peru were also able to recover from the diseases and mass migrations caused by the plagues. Old World technology were at first adopted reluctantly, then enthusiastically and then innovated by the peoples of the New. And soon, the New World could match the Old in technology.
The Inca and Mexica peoples would much later form empires that could match the empires of the Old World in grandeur.
In the North, the Gauls formed an hegemony which was overran by later Germanic migrations to some extent. Speaking of the Germanics, the Norse after an age of activity of plunder and terror, settled down and were able to settle the OTL St Lawrence Valley around the time Carthaginian settlements grew to wealthy trade ports and merchant towns in the Caribbean and New World coasts.
In the East, the Warring States Period in China arrived to a similar conclusion as in OTL. The Kingdom of Qin created a short-lived empire and then fell in a civil war before reuniting again into a more moderate and long-lasting dynasty. This dynasty, the Chu, took the place of the Han in OTL and fell apart in the 3rd century AD. Centuries of division followed by unification into a grand and cosmopolitan Zhou China which brought the glories of Chinese civilization to Nippon, among others and pacified the northern barbarians. However, various nomadic peoples, in succession, conquered much of China once the Zhou dynasty declines and falls due to internal issues. This is why much of the West calls China Cathay in this world - named after the Khitai who conquer China in numerous occasions.
India following the Maurya was initially very similar to OTL. Various regional kingdoms dominated the political landscape while occasionally one kingdom manages to claim hegemony like the OTL Gupta. However, these rarely last more than a few decades and a long "Warring States Era" in India develop. Fate would decree, however, that this series of events would not last forever, though, and by about AD 800 one single dynasty based in the populous Ganges valley conquers all of India in a much more definitive manner than what the Maurya ever achieved. This was hard fought and what resulted ensured that the previous state of affairs could no longer continue. Thus, they - the Neo-Gupta - usher in the Imperial Age for India. The Neo-Gupta were patrons of Buddhism which inevitably flourishes and eclipses Hinduism as the dominant religion of India. The peoples of Southeast Asia are brought into an Indian-style tributary system thanks to the efforts of the Neo-Gupta predecessors - the Chola and other Tamil states - whose efforts were taken over by the Neo-Gupta.
Once this Neo-Gupta Indian Empire declines and falls, they are replaced by the energetic Satavahana who are dominant especially during the 13th and 14th centuries.
Africa develops her own states and imperial traditions. Aksum in the east and Timbuktu in the west form the core of these empires. Though it must be mentioned that the Kongo in central Africa are a notable regional power in their own right. And of course, Kilwa, the queen of the Swahili cities, looks overseas to expand her influence.
In the lands once under Macedonia however, the region was afflicted by numerous warlords and shifting regional powers for a few centuries. It seemed as if Macedonia was only to be found in the pages of history, forever lost to the world. But, the Macedonian imperial idea proved too strong to resist. Soon enough, an ambitious warlord fulfilled the dream of centuries and was able to conquer his rivals and reunite the lands once part of the old empire. Centuries have passed since Macedonia was a great empire on earth, and in a similar manner to OTL China, Macedonia long divided has united. The Magi tell us only Helios-Ahura Mazda, who gives to each empire its allotment, knows how long this iteration of Macedonia will last.