"Should Batman kill the Joker?" is a perennial favorite among superhero fan conversation topics, always leading to a variety of different answers. A Golden Age appearance aside, Batman's bosses at DC Comics have always answered the question the same way, however: Hell no.
Part of the reason for that is practical. You don't kill off a popular, money-making character (well, you can now and then if it will make more money, but then you have to bring the character back to life somehow). Part of it is smart franchise management. If Batman kills off his enemies, then he runs out of guys to fight awfully quickly. There's a reason Spider-Man has such a big and colorful rogue's gallery to fill movies, cartoon and toy lines with, while The Punisher doesn't. But a big part of it has to do with Batman's characterization. Maybe it doesn't make sense to not kill a mass-murderer you find yourself in deadly combat with on a bi-monthly basis, and sure, it makes even less sense to go out of your way to save the life of said mass-murderer as Batman regularly does for The Joker and his other foes, but then, dressing up as a bat to fight crime doesn't make much sense either—Batman's weird, and that's what makes him so appealing. Of course his moral code is weird too.
The red, un-crossable line Batman has drawn between beating someone within an inch of their life and actually killing them is one shared by most superheroes. The hero pushed to the limit finally getting the villain at their mercy at the climax and forced to decide whether or not to end the villain's life of evil once and for all is a staple of super-comics.
And it hasn't changed all that much in the years since, say, "The Trial of The Flash." Particularly in the DC Universe (The Marvel heroes embraced killing foes en masse during 2008's Secret Invasion, in which they went to war with the alien Skrulls).
Wonder Woman famously killed Max Lord in the heat of battle in 2005, and it lead to year's worth of stories in which her fellow heroes debated with her over whether or not it was a just act. She was even on trial in the international criminal court for a while.
Green Arrow killed the villain Prometheus at the climax of controversial 2009-2010 series Justice League: Cry For Justice, and it lead to a weird storyline in which his fellow Leaguers tried to bust him and he ended up being exiled form his hometown.
And during the 2009 story arc that introduced the Red Lanterns in Green Lantern, space policeman Hal Jordan wrestled with the idea of his bosses executing his mentor-turned-archenemy Sinestro.
As you may have heard, the DC Universe has changed quite a bit since then.
I haven't been reading very many of "The New 52" books as they've been released. In fact, I can count the number I've been reading on one hand: Justice League, Justice League Dark, Aquaman, Green Lantern and Wonder Woman. But even among that very small sampling, I've noticed a trend emerging.
In the pages of Aquaman #2, by Geoff Johns, Ivan Reis and Joe Prado, Aquaman and Mera confront the humanoid creatures called The Trench:
In last week's Green Lantern #3, by Johns, Doug Mahnke and a whole mess of inkers, Jordan is confronted my a member of the yellow Sinestro Corps, and deals with him thusly:
And in this week's Justice League #3, once more written by Johns and drawn by Jim Lee and Scott Williams, the various heroes of the Justice League confront Darkseid's armies of parademons. I was a little surprised to see such an aggressive, ruthless Superman doling out pain to his foes:
But I was more surprised still to see this image, in which Superman takes off a Parademon's arm and head:
Even the new sword-wielding Wonder Woman, who chose a maiming blade over a capturing magic lasso in her first appearance and, in this particular issue, chops limbs off like she was chopping vegetables, doesn't take anyone's head off in this issue.
I suppose the argument could be made that these aren't human lives. The Trench are humanoid, though, and speak, so they're clearly a bit more evolved than animals. That yellow Lantern is an alien instead of a human, but so too are all Lanterns save a handful (and hey, Superman's an alien too, and maybe Mera, or are people from different dimensions more ultraterrestrial than extraterrestrial...?). I'm not sure how "alive" the new Parademons are, and they certainly have a lot of mechanical and/or robotic looking bits to them, but they also bleed, and I didn't see Superman X-Raying their chests before ripping them apart, or Aquaman checking one's pulse before shoving his trident into its back.
Basically, the New 52 Joker might want to lay low until the New 52 Justice League work through their collective issues.