You can activate the 1999 mode cheat in Bioshock Infinite by pressing the following at the title screen:

Up, Up, Down, Down, Left, Right, Left, Right, B, A.

1999 mode makes the game very difficult and each time you die it will have "severe consequences" (be sure to comment below and tell us what that means exactly!). In fact if you die without the correct resources, your game will be over and you'll need to start from scratch!