A few people have asked me to recommend comic books for them after reading Watchmen, which is kind of like asking someone to recommend movies for you after you've seen Casablanca. The question I always have to ask is, "well, what do you like?" If you're willing to do a little research, there's a comic book out there for you, and I don't just mean superheroes. There's Westerns, science fiction, crime drama, comedy, horror, autobiographies, romance, whatever you're interested in. And if you're in an international mood, you can find comics from the UK, France, Italy, Korea, Japan (lots and lots of stuff from Japan), and even far-away Canada!
After the so-so performance of Watchmen at the box office, DC Comics decided to release a list of comics aimed at older readers. You can look at their fancy website here. There are three problems with this list: it's too superhero-centric, it contains a lot of books that aren't actually good, and it only has books published by DC Comics (obviously).
But for those interested in getting their feet wet in comics, the After Watchmen list is as good a starting point as any. After trimming the fat, here's what I would put forth:
Saga of the Swamp Thing: Alan Moore is pretty much the only writer who ever did anything worthwhile with the character of Swamp Thing.
The Dark Knight Returns: Before Frank Miller became a walking parody of himself, he wrote really entertaining, noir-ish stories. This title effectively redefined Batman for an entire generation. You can see its influence in both the Tim Burton and Christopher Nolan Batman films.
All-Star Superman: a recent work by Grant Morrison, it comes close to being the Platonic ideal of a Superman comic, and for one brief instant I understood why some people really like the Big Blue Boy Scout.
Animal Man: an older work by Morrison in the same deconstructionist vein as Watchmen. It also touches upon animal rights and violence in mainstream entertainment.
V for Vendetta: don't let the movie fool you, this book by Alan Moore was really good (and very British). This also holds true for From Hell and League of Extraordinary Gentlemen. Actually, anything by Moore is probably worth checking out.
Transmetropolitan: don't care about superheroes? What about a story by Warren Ellis where a crazed, junkie journalist (essentially Hunter S. Thompson) fights a corrupt government in the future?
Y: The Last Man: High concept pitch by Brian K. Vaughn: every mammal with a Y chromosome dies one day, except for one wannabe illusionist and his pet monkey. Hijinks ensue.
Sandman: Neil Gaiman's massive dark fantasy about the Lord of Dreams and the art of storytelling.
Preacher: Did you ever feel like punching God in the face? Well, so did Garth Ennis, and this book reads like one long "fuck you" to Christianity, but along the way it tells an exciting adventure story with richly drawn-out characters.
This list isn't meant to be exhaustive, and there are other titles on the After Watchmen site that are worth checking out if these books get you interested. I'd also recommend looking at the websites for smaller publishers like Dark Horse, Image, IDW, Dynamite, Oni, Fantagraphics, and Drawn & Quarterly. And of course, there's Marvel if you want more superheroes.