I am rather confused about the ending.
Why does Booker getting drowned at the baptism prevent alternate versions of him from becoming Comstock? Just because the game says so? Wouldn't there be alternate universes where he did not get drowned and still went on to start Columbia?
And why does the game hint that there is some alternate universe where he and Anna live happily ever after? If the drowning effectively ended his timeline across all universes, wouldn't that be impossible?
Taken from a gamefaqs post that explains it well I think.
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Bioshock Infinite is a story, ultimately, about saving the universe. This only becomes clear in the last two scenes, when Booker is drowned, and has his child post-credits. Let's rewind.
The universe is one of constants and variables. It is not chaos theory. There are not universes born of every choice. Every universe reaches certain key events. Constants, and variables. The major constant that is addressed in the game is that of the baptism. Seemingly, we have two outcomes: one of Comstock, and one of Booker. But there are four. Four possible outcomes that serve as anchor points for the multiverse, with minor variations of these branching to and fro.
The first: Comstock is born, rules Columbia. Elizabeth is abandoned, and goes on the destroy New York.
The second: Comstock is born, but Elizabeth is saved, and Columbia is destroyed.
The third: Booker refuses baptism, but sells Anna to settle a debt to Comstock, who has breached his world to engineer this manipulation, and launching the whole adventure.
The fourth: Booker does not sell Anna, because Comstock never breaches his world.
The first three universes have one thing in common: the Lutece Tear. A rip in spacetime made possible by funding from Comstock. The first three outcomes are reliant on an existence of Comstock in order for the tear to be made possible. The tear is the real problem. It is causing the universe to fall apart. It is evident by the tears that exist around Columbia. And the problem is exacerbated as Booker and Elizabeth jump from world to world, resulting in people being caught between dimensions. The very fabric of spacetime is threatened. The only resolution is to eliminate the possibility of the tear. Lutece will always exist. But the funding does not without Comstock.
When Booker drowns, he closes the loop, and eliminates the possibility of Comstock. He rewrites the universal constant. The question is no longer "does Booker accept baptism?" There is no question. Every universe that is influenced by Comstock is eliminated, because there is no Comstock. This leaves only the fourth possibility, where Booker always refuses baptism. He never sells Anna. A new constant, no longer a point of divisiveness in the universe. There is no more tearing, no people caught between dimensions. The universe is fixed. No plot holes, no paradoxes. Everything wrapped up nice and neat.