In general, my musical taste is pretty diverse, but I'm quite demanding. In most cases, I'm only interested in music that is fairly complex. That can be almost anything however, ranging from classical music over jazz fusion and prog rock to heavy metal (and beyond). To round things off, examples:
Cirith Ungol - King Of The Dead - Very obscure heavy/doom metal band from the 80s, with an amazing guitarist and possibly the weirdest singer on earth
Iron Maiden - To Tame a Land - I rediscovered this just yesterday when listening to "Piece of Mind", and I was utterly amazing even though I already knew the song through and through. Lyrics based on Frank Herbert's novel "Dune", long instrumental passages that quote classical music etc. (Live version because Sony and Warner Bros. are being bitches on German YouTube)
Return To Forever - The Sorceress - None of you probably heard of Chick Corea, Lenny White, Stanley Clarke and AlDiMeola, but if you were into jazz, they'd be your Gods. Very funky, extremely complex and utterly beautiful. AGain, live performance, but this time because this version is utterly amazing.
Sonic Youth - Unmade Bed - Sonic Youth are basically indie rock heroes. They use weird guitar tuning, lots of experimental noise and three guitars playing trippy melodies at the same time. This one is rather calm, but stunning.
Pearl Jam - Red Mosquito - Possibly the least complex thing I linked here, but still far from your usual rock music. You might know Pearl Jam as a Grunge band, but that's not really accurate.
Carl Orff - O Fortuna - Please don't go all "Ooh, I know that" at the beginning, but listen to the overlooked second half.
Rush - Xanadu - Rush are amazing, end of story. Penopat made a level called "La Villa Strangiato", whose name comes from another great song of theirs.
Okay, those are quite a few examples, I tried to be as diverse as possible. That said, I could now rave on how I hate almost everything anyone else linked here, because I find it shallow, boring etc., but I don't wanna annoy you.
EDIT:
Scatman John, Daft Punk, Linkin Park, Lady Gaga, Justin Beiber (lol), and...
Was that supposed to be a diverse selection? If so, it's not. Also, why do you people keep misspelling Bieber?