BioShock Infinite is very much a BioShock game. That’s obvious from the title, of course. But the two games are more intrinsically linked than that. Characters, mechanics, and settings are all shimmering reflections off the same pool of water: the final image is slightly different, but both come from the same root picture. BioShock and BioShock Infinite are in a quantum entanglement, stemming from the same strand of DNA. As a result, they share many ideas.
In the second article, Lachlan Williams explores some of the more mechanical similarities, and how they express various themes that link the two games.
Again, this article contains SPOILERS for BioShock and BioShock Infinite, so do not read if you do not wish to be thoroughly spoiled. Ye have been warned. Again.