Okay, maybe Bruce Wayne hasn't always been depicted as a loon, but he certainly often has. Especially in the last couple of decades, when Bats was ground zero for the move to "deconstruct" superheroes. Alan Moore had Batman sharing a demented laugh with the Joker at the end of The Killing Joke. Alan Grant and other writers played on the idea that Batman was as disturbed as the baddies he hunted - and Grant even had that Skinnerian psychotherapist, Arkham, lock Bats up in Arkham Asylum with the rest of the crazies. More recently, Grant Morrison had Batman creating a fantasy "backup personality" for himself in case he ever got drugged and driven over the edge. A crazy plan for a crazy situation, or just the ultimate proof that Bats dances over the edge? Perhaps most tellingly, back in the late 1980s, Steve Engelhart had Batman utter the classic line, "My world goes crazy sometimes, but I don't." Which is like saying, "I'm a nut, but I'm in denial about it." It's okay, Bats.
The write-up does include Batman's first stint in Arkham, but it's a crime not to mention "Arkham: A Serious House on Serious Earth," written by Grant Morrison and illustrated by Dave McKean (famous for his Sandman cover artwork). In "Arkham," Bats has to enter Arkham after a full-scale riot at the request of the Joker. Bats' take on it? "I'm afraid that when I enter that place....it will be like coming home."
Frank Miller also tees off on this in "The Dark Knight Returns." In that non-canon run, Batman's retirement renders the Joker completely catatonic and harmless. The second that Batman is back, though, the Joker becomes his old, mass-murdering, crazier-than-any-character-in-anything self. (SPOILER ALERT: just to celebrate Batman being back, Mr. J poisons some cotton candy and hands it out at the fair, murdering dozens of cub scouts among others. Jesus.)
POST SCRIPT: It had slipped my minds, but speaking of "elseworlds"-esque storylines, it's worth noting that in Mark Millar's "Red Son," which imagines what would happen if a 12-hour difference had landed Kal-El in the USSR rather than the USA. In this alternate reality, Batman is a crazed, barely-sane anarchist terrorist who delights in messing with Superman's perfect Communist dys/utopia. He ends his illustrious career as (SPOILER!) a suicide bomber. Even in a parallel reality, Batman is nuts. Behold: