There will obviously be endgame spoilers for Bioshock Infinite in this thread, so go away if you don't want to know what happens.
The title should not be considered a spoiler since we know Booker doesn't go to New York 1/4 of the way into the game. If anyone feels like this is still too much, I'll change it. Another warning, spoilers below.
[spoiler]Robert Lutece hired Booker to get Anna for Comstock (was the relationship between Booker's debt and Lutece ever explained?), and then hired him again to rescue Elizabeth from Comstock. Presumably he thought he would get her out of Columbia and it'd be over...but obviously that didn't happen.
So, if Booker had managed to find an airship to get away, and tricked Elizabeth into thinking they were going to Paris. Once he arrived at New York, who would he go to? Who was actually in charge of Booker's debt? This really bothers me, since it's a huge part of the plot. Did Robert Lutece work for the Pinkertons? Did he lie to Booker about wiping the debt away?
I may have missed some voxophones on this, but I feel like we're missing some key details about the beginning of the game.[/spoiler]
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Booker wouldn't, won't, and didn't make it to New York. It is a constant in all of the parallel realities. As for the debts, it is a false memory. This is explained twice in the game, once at the beginning when Lutece says "I hope this experiment pays off.", and once at the end when Lutece says "See his mind is putting his story together with memorys that aren't there." (Don't qoute me on that) It was just a false memory used to get Booker to go get Elizabeth and set in motion the events that took place. I wrote a thread about it if you want to check it out.