Roman Polanski’s The Tenant
Bob Rafelson’s Head, starring the Monkees(!)
Scorcese’s After Hours and/or The King of Comedy
Miracle Mile
Donald Cammel/Nic Roeg’s Performance
Some of these may be hard to find, but deep weirdness is assured.
I would throw in these two gems that messed with my mind a little:
House of Yes - starring Parker Posey
The Spanish Prisonner - by David Mamet
How about Heist. I liked it because you never know when someone is faking or lying. And you don’t know who is on who’s side.
As for movies with strange endings: The Others. I didn’t like the movie at first, but the ending was very good and fits into the Vanilla Sky/Sixth Sense type. You could go back interperet many parts of the movie differently after you see the ending(like the husband, the ghost, etc.)
Also how about Full Metal Jacket and Apocalypse Now (I think someone already mentioned it.) You go into both of them thinking you’ll see another war movie, but they have much deeper meanings.
After I saw Un Chien Andalou, I spent a bit thinking about it, analyzing what it meant. Then I threw up all over myself.
Anything by Pedro Almodovar. The Bad Lieutenant. ** I second the already mentioned The Cook, The Thief, His Wife and Her Lover and ** Run Lola Run. No one’s mentioned American Beauty yet?
American Beauty, I would have to add that one. ExisTenZ of course, too. Altered States, too. Anything by Todd Solodz especially. And no one’s mentioned The Sweet Hereafter yet?
Dark City was flawed, but it did have a lot of food for thought. I’m still digesting Bowling for Columbine, but maybe documentaries are outside the scope of this thread?
The real mind-shifts I’ve had have mostly been induced by anime-series – especially Neon Genesis Evangelion and Serial Experiments Lain. Again, probably outside the scope of this thread.
Louis Malle’s Black Moon
Finishing off my Top 10
- Repo Man
- The Ring
- Liquid Sky
Tetsuo: The Iron Man, 1992, directed by Shinya
Tsukamoto.
Weird and disturbing.
But not as weird and disturbing as Tetsuo II
Arizona Dream, a 3 hour surreal comedy starring Johnny Depp and Jerry Lewis.
Cemetery Man, a surreal zombie movie starring Rupert Everett.
O Lucky Man, Malcolm McDowel as a coffee salesman traveling thru several loosely connected dream worlds. With a cool 70s rock band as a Greek chorus.
Santa Sangre, a very surreal take on the slasher flick with circus imagery and dead people turning into swans.
For Y’ur Height Only, a Philipino midget spy movie- quite possibly the funniest thing ever filmed.
Wild Zero, Japanese punk rockers vs. zombies and aliens, along with a touching love story.
For a change of pace, you could try some weird quirky comedies that are “cult classics” and require repeated viewings to absorb every odd detail. Two examples that I’ve seen recently are: Buckaroo Banzai and Rustler’s Rhapsody.
A favorite line from Buckaroo Banzai—“I’ve been ionized but I’m OK now.” (And John Lithgow’s performance is fantastic.) “Rustler’s Rhapsody” has a hilarious and unexpected performance by Andy Griffith, and the line about “confident heterosexual” is pretty damned funny.
Both these films are outside of the norm, that’s for sure.
Solaris
I liked the film, but the experience was hampered by a companion who had a bad attitude about the whole movie before we even stepped into the theatre, and would not shut up about how “crappy” the film was. I got very irritated with him very fast. The film was good. I was a tad disapointed with the lack of action you expect to see from James Cameron, but oh well.
To add:
Donnie Darko.
Ghost World.
Taxi Driver
The Virgin Suicides
BTW, you should also see the original Solaris
I’m going to try not to double any of the suggestions above, and not even mention those of which I didn’t like 
Happiness
Lulu on the Bridge
the Linguine Incident
Double Happiness
Love and a .45
Titus (the 1999 version)
Life In The Fast Lane
Skeletons in the Closet (horror, sort of)
Doom Generation/ Nowhere (the second is somewhat less disturbing)
Koyaanisqatsi
Lost Highway
Two words: Ken Russell.
I don’t know much about this director, but I don’t envy his shrink.
The City of Lost Children: A 1994 French film about a man who kidnaps children in order to steal their dreams, because his lack of dreams is making him old. Has Siamese Twins, four or five clones, a midget wife, a brain in a jar, some orphans, and a circus strongman.
Ew, I didn’t know there was a sequel. I am both intrigued and apprehensive.
There are a number of good films listed in this thread. I think I’ll print this out for the next time I hit the video store.
Shinya Tsukamoto, the same guy who made Tetsuo and Tetsuo II (which I found somewhat dissapointing) also made the movie Tokyo Fist. His other movies are harder to find, but Gemini, based on an Edogawa Rampo story, is quite good- a man is replaced by his doppelganger.
The lead actress, set designer and costumer for Tetsuo, Kei Fujiwara, made the movie Organ, which was very interesting. It was a really disturbing movie involving Yakuza harvesting people’s organs and a screwed up family that runs the racket. one of the most notable sequences involves a naked woman turning into a moth. Very disturbing, and you can definitely see the influence of Tsukamoto.
Forbidden Zone was pretty damn weird. Hervé Villechaize as the king of an underground kingdom? Oingo Boingo’s Danny Elfman as Satan? The movie also featured simulated sex, musical scenes and really fake looking monster mutilations (i.e., mutilating really fake looking monsters).
This movie is not only really bizarre, it also shows the beginning of Oingo Boingo and the beginning of Elfman’s very successful career as film composer.