Does anybody else feel like Jaime Lannister is being made into a more sympathetic character? I'm not sure if losing his hand and his travels with Brienne were meant to actually improve him as a person, but some of us will never forget that he pushed Bran Stark out of a window. 
Yes, it happens in the books too. The same can really be said for the Hound - he killed a little boy too, but a lot of people like him. You have to remember that the world in question is very much a
crapsack world, and as such it will take a serious dump on people who would otherwise be rather decent individuals. That can make some of them quite jaded.
Jaime is sympathetic as a character because he's not an irredeemable monster like Joffrey, Gregor Clegane, or Ramsey Snow, and as such his motivations are much more human. He has a sense of honor, but he's got other values as well, and the world has forced him to choose between his honor and those other things too many times. It's something a normal human being can empathize with.
Also, he changes as the story goes on. Maybe gets a little less jaded, maybe tries to redeem his honor a bit. And change is what all but one of the book's perspective characters really go through, and we all know what happened to him.
