Now we've all seen the trailer and posters we have a bunch of theories but very little when it comes to hard facts.
So in that spirit I will try to keep my theory as down as possible and if possible in accord with what we know about Infinite Multiverse.
Elizabeth/Anna and Booker are they the same as the ones we know?
Now most of us expect this to be sort of continuation of Columbias story. However I have to ask why?
We know Booker has AD on his hand correct? And We know that this Elizabeth is looking exactly same as ours Elizabeth except for clothes and haircut. Some say she is older but I have to ask myself if that isn't merely because of different haircut and more mature makeup and clothes.
You know opinion on how should girls in their late teens/early twenties dress was very different in 1912 and 1957. Not to mention the highly religious population of Columbia probably would have very different opinion on the matter than highly secular society of Rapture.
OF course the AD is still unexplained. But think about what Liz says during epilog: ..here's always a lighthouse, there's always a man, there's always a city. Or something like that.
There are INFINITE Columbias and therefore INFINITE possibilities, it is not that hard to imagine that in some Rapture happened story similar to that of Columbia.
What if Booker we find in Rapture sold his daughter for some sort of ADAM related testing? He then carved AD on his hand and continued to live as private investigator.
Of course there is a matter of Elizabeths age. But Rapture had technology to alter growth and maturation of human body so that is not an issue.
ADAM testing perhaps something related to teleportation plasmid would allow for something like tears so the core mechanics of Infinite could be kept.
Of course there is a possibility that KL simply transported our Booker and Liz into new setting and that he's going to justify it by some syphon related stuff.
But in my opinion that would be too predictable and let's face it how many of us new BEFORE Infinites release how it would end.
Anyway this is my theory unsupported by facts. I hope I didn't bore you much.