I remember playing the original Bioshock a few years ago on Xbox 360 and really liking it. The sequel was originally done by a different team but I loved that one too. Bioshock Infinite is done by the original team, but set in a new city with original characters. I ended up getting this one on PS3 which doesn't matter to me since it's virtually the same as the Xbox 360 version.
Instead of the under-water dystopian city of Rapture, this game has you in the floating city of the skies known as Columbia. You play as Booker Dewitt, a man who is a bit of trouble but is offered a clean slate if he rescues a woman named Elizabeth. The game plays a lot like the original where you have regular weapons with your right hand, and almost-magical abilities with your left. It plays less like a traditional FPS but more like an adventure game.
The thing that stands out the most is the character Elizabeth. Saving her is easy, getting her out of Columbia is not. She can't die in the game (so there's no need to worry about her in battle), and she has magical powers to open up portals. Which can lead to either new areas or bring in items or robotic allies to your side. She also can throw ammo, health, and other items when you're running low during a tough fight.
I can't say that I enjoyed Bioshock Infinite more than the original, but I was very pleased with the game. I really liked the dystopian setting of the game and how it's a parody of theological culture and right-wing political nationalism. The story is good but I expected a twist near the end (like the original), and it's rather strange but very creative. I was honestly surprised by how cool Elizabeth is, she's a vital ally in combat and certainly not much of a damsel in distress.