What happens in the ending of Batman Arkham City?
Solution 1:
I'm going to go out on a limb here and say, no, he doesn't.
The Joker's such an iconic Batman villan, there's no way to kill him. The Joker's an integral part of Batman's universe. As with every comic book character, he can appear to die and then show up in the next issue/episode/sequel as if nothing has happened.
I would like to note, however, that this game was Mark Hamill's last performance as the Joker.