The Bear Flag Republic never had a chance of staying independent. It was founded by, and led by Americans including an active duty American military officer for the express purpose of gaining the territory for the United States. Now if the Californios revolted during the Mex-Am War and gained independence, then it will proceed much like Wolfhound described, where California will become American sooner or later due to immigration/goldrush unless it gains a foreign backer. Britain is the only plausible foreign backer, but without massive PODs, I don't see Britain as being interested.
Exactly my thoughts, although if one imagined an early PoD, in which a Californian independence movement arose among the indigenous Latino population at roughly the same time as the Mexican revolution on 1821 and the state acheived independence as a non-anglo nation in the early 19th century, this might affect later history. For one thing, California and British North America might establish a common border on the Pacific in the 1820's or 1830's, precluding any US claims on Oregon.
A major purpose of the Mexican-American war was to gobble up all of northern Mexico, especially California and reach the Pacific. If California was not part of Mexico, but another latin american republic, this complicates things. Even if a war of conquest with Mexico was justified on the basis of the Texas border disputes, what would be the reason to delare war on California when the dispute was supposedly with Mexico? I could see a situation where the US assimilates the former Mexican territories that become New Mexico, Arizona, Nevada, etc., in the 1840's-50's but westward expansion temporarily stops there as the American Civil War puts further expansion on hold for a decade or two. In Through skillfull diplomacy California might prevail on Britain to offer some alliance, although whether or not Britain would want to risk war to keep the US from having a Pacific coast would need to be explained.
However, none of this means an independent hispanic California would end up with the world's 9th largest economy in 2009. This never happened anywhere else in Latin America and it's hard to imagine that, without the influx of American settlers and capital, California would be anything other than a much smaller, less populated, and more impovershed version of Mexico.
