After the wounded knee Booker went into depression and started drinking and gambling heavily. He went to the baptism, but didn't go through with it and continued gambling and drinking. He then had Anna/Elizabeth with his wife who died at childbirth. Booker in the other world did get baptized and took the name Zachary Hale Comstock. He became radical cult leader and with the help of Rosalind Lutece created Columbia.

But due to extended exposure to the lutece device he became impotent and aged quickly. He however had this ''vision'' that his daughter would burn down New York, so he had to get one somehow.

Now back to Booker. At this point He had severe debt problems. He was contacted by Robert Lutece who had been communicating with himself from the 'Comstock world', Rosalind, to give Anna to Comstock and get his debts paid in exchange. Booker agreed to this but regretted his decision and tried to get Anna back, which led to him and Comstock fighting over her and her losing pinky.

Now during next twenty years Comstock got his wife killed and tried to get Luteces killed so that no one knew the origin of the girl. Booker in the other world had gone even deeper into depression after losing Anna and gambled his money away for next twenty years.

In 1912 Luteces came to Booker offering him chance to go rescue Anna from Comstock, to which he agreed. However the step through worlds resulted in him losing his memories and creating new ones where he was hired to get the girl for his debts to be paid.

All credit for this text goes to /u/juhinaattori Source: https://www.reddit.com/r/Bioshock/comments/58szr9/comment/d9308ua?st=IVBGP1I7&sh=bdcb9ceb