Bioshock Infinity is like Looper
When you think about it, it makes sense.
Massive spoilers:
If Booker refuses the baptism, he will become an alcoholic gambler who will sell his daughter in order to settle his debt. The man who will sell his daughter to will use her as fuel for a flying city that would destroy the world.
If Old Joe goes back in time to kill a telepathic kid, the Rainmaker will never exist and his wife would be safe from murder.
However, if Booker accepts the baptism, he will recreate himself and giving himself a new name, Zachary Comstock, becoming Prophet of the flying city, kidnap Booker's daughter from an alternate reality, then destroy the world.
However, if Old Joe doesn't kill the psychic kid, the kid will grow up to be the Rainmaker to kill all loopers and rule the world.
Solution: For Booker, he drowned himself so that two versions; one who will sell his own daughter to pay off gambling debts and the other who will become deeply religious to destroy the world with the flying city will never happen. For Joe, he shot himself in the chest so that his future self never existed to kill the kid's mother and turning him into the monster that he is trying to kill.
It does make sense of trying to off self to kill other self and preventing alternate timelines.
