What are some very weird/avant garde but really good movies?

The Virgin Suicides (1999)

Fat Girl (2001)

Fargo (1996)

Natural Born Killers (1994)

Apocalypse Now (1979)

Magnolia (1999)

As well as other movies from the directors of the films you mentioned, (Southland Tales) (Julien Donkey-Boy, Mister Lonley) ("Six Feet Under") (Resevior Dogs, Kill Bill) (2001, Dr. Strangelove, Lolita, Eyes Wide Shut) (Pi, Requim for a Dream).

Hope that's helpful

Happy Endings

Children of Men

Pi (same guy who did The Fountain and Requiem For A Dream)

Oldboy (a really good korean film)

Altered States

Slacker

Waking Life

try looking these up on wikipedia or IMDB.com to get a better sense of what they're about.

How about Girl Interrupted, American History X, Poison Ivy, Super bad, Donny Brasco.

Try a couple of those.

try Art School Confidential, Tideland, Mirrormask (come on Neal Geiman and the Jim Henson company), Pans Labyrinth, or The Science of Sleep

edit:

I like Mr. Ee's answers as well.. (although, I wasn't really thrilled with Happy Endings..it was interesting)

J-Men Forever

The Abominable Dr Phibes (Old flick starring Vincent Price)

Paprika

fight club (i consider this both weird and avant guarde; oldie but goodie), requiem for a dream, sleepers, pi, momento (great one), kill bill 1&2, what dreams may come.

i love odd movies, but i can't think of anymore that i love.