@Višeslav I do not think that Alexander was someone eager to put up with defeat... So reorganize forces and then push east, further then before, to Ganges, to Pataliputra, to Bengal bay!
Would then greek world be more affected by buddhism?
That's true. What I meant is he might not bother with some of the less profitable areas in the south (or alternatively would die trying to take them). He might leave them for later as he conquered more useful lands to the west, eventually dying. I think the Ganges valley sounds like a reasonable focus for his expansion. Elephants would not be a problem for him, as he had fought them before and had quite a few of his own. His army was better equipped and larger than those of most Indian kingdoms at the time. Not insanely so, but a bit, so this campaign should be ok for him if he lives longer.
He faced odds so much worse in Persia and won, so I don't doubt India is possible assuming he lives long enough to do it.
However, transport and logistics are a big problem, and morale. His soldiers weren't all too happy with the forced march across the entire known world. India's big, and, unless you take it in chunks (which was not exactly Alexander's style, it seems), remaining able to bring supplies over as you expand is difficult, then again, big Al's army was pretty good at foraging, and specifically campaigned along food-rich routes or took detours to get it. Delivering more armor (which they actually did do according to some sources) would be harder and harder as they got further from their territory, but, again, they faced worse odds in Persia.
We do need to note, also, that if Alexander gets older, he might become less hell-bent on conquering an entire continent. Most leaders like him either die young, lose and get imprisoned while they're still young, or calm down a bit as they age.
I'd expect the maximum extent of his empire would be conquering India north of the Deccan Plateau (roughly), Carthage (so a huge chunk of the Mediterranean), what he already had OFC, maybe de jure controlling northern Arabia (he wanted to march his army across it just to assert his claim to it, though the soldiers might be a little disgruntled after that), de facto, I'd expect local Bedouins would pay an annual tribute or something, and then finally, vassalizing Nabatea (Sinai and the bits of modern Jordan and Saudi nearby), Gerrha (roughly Al-Hassa, Bahrain, and Qatar) and Mazun/Makkan (Omani & UA coasts, historically aligned with whoever rules Persia). He might also vassalize Sabba, Hadramout, and Qataban (Coastal Yemen and western Oman more or less). The other Greek states (like Massalia (marseille) and the Black sea kingdoms) for sure, and possibly, thought it's kind of unnecessary at that point, the Etruscan league. Keep in mind, this behemoth of an empire is the absolute maximum.
As for Buddhism, the Hellenistic states IOTL were in consistent contact with India through Bactria, a Greek Buddhist kingdom, but did not pick it up. Assuming Alexander lives longer and establishes a stable line of succession, a more unified empire would allow Buddhism to spread west much more easily.
Seeing as the movement would be centered in a region within the empire, Buddhist thinkers might visit the court of Alexander or one of his successors, maybe the Buddha would be viewed more as a philosopher than a religious leader by the Greeks though. Stoicism was pretty popular at the time, and Diogenes the cynic was greatly admired by Alexander, and these had quite a bit in common with Buddhist teachings (Buddhism might be seen as a kind of pacifist stoicism). This means that, as a philosophy, it is likely to be pretty popular, with, I'd expect, some religious syncretism with whatever ends up being the dominant religion in the western part of the empire (likely still Hellenic Polytheism, though other religions might become popular there as well).
If the empire does break before the Buddhists have a chance to spread throughout it, expect it to be fairly widespread in Indian and Central Asian centered successor states, but a Persian-based successor state might find Zoroastrianism more appealing simply because it was already widespread there, but the Buddhists could become a significant minority. Buddhist-Zoroastrian syncretism is also plausible.
Regardless of how popular it is, Buddhist thinking will certainly have some influence of the future of Hellenistic society, and philosophy ITTL.
That's just my take though, I'm no expert. Sorry for the really long post that isn't too related to proposals and war aims (though the majority is about Alexander's future war plans, so I guess it still counts)
