You don’t like Gattaca but you like Thelma and Lousie?
Strange but to each their own.
I’ll add
Babe the Gallant Pig
and
Project X
If you liked the Matrix, give Dark City a try. It came first and has a similar premise.
Yay for all these, especially Mann and Sirk.
An interesting film, though not Great with a capital G like the ones in ArchiveGuy’s list, and one whose approach is more literal and concrete, is Mindwalk. Sam Waterston plays an American politician–candidate for the presidency, IIRC, on vacation in France. He’s visiting John Heard, an old college friend, who’s a poet. They visit Liv Ullman, an artist IIRC, and walk and talk. Some fascinating issues are discussed and dissected, and it actually manages not to get boring. I don’t recommend this film to everyone, but based on the OP, I’d recommend it highly to Achernar.
I must also second the recommendation for Altered States. Again, not a Great film, but like another film scripted by Paddy Chayefsky, Network, manages to be big-time entertaining while delving into some very “deep” issues.
Paris Texas is Wim Wenders, not Jarmusch. But this reminds me of two of my favorite films, both relevant to this thread: Jarmusch’s Dead Man, and Wenders’s Until the End of the World.
Dead Man is a brilliant, haunting film about the journey each of us is taking toward death. Amazing performance by Johnny Depp as an accountant named William Blake, though Crispin Glover might be my favorite thing about this movie. Unless Robert Mitchum’s final appearance in a film is my favorite thing. No, maybe it’s Iggy Pop as a frontier transvestite outlaw. Or the Indian named Nobody.
Wenders’s epic Until the End of the World is an epic near-future SF epic; did I mention it was an epic? It’s filmed on location pretty nearly everywhere in the world. It’s a weird pomo-noir globetrotting cautionary tale, as only Wenders can do. The beautiful Solveig Dommartin follows William Hurt areound the world He’s on the run from the CIA, who want to steal his father’s invention that records dreams and vision. He invented this so that his wife, Hurt’s mother, who is blind, could see what others see. Meanwhile, there’s a satellit hurtling toward earth, great explosions are expected, etc. Dommartin and Hurt flit through NY, Paris, Tokyo, China, etc., and fianally end up in a classic mad scientist’s lair underneath the Australian Outback, with Max VonSydow as Hurt’s father. This film covers a lot of ground, but it keeps you fascinated, and leads you along some interesting intellectual paths. Plus it has a GREAT soundtrack.
The Deep
The Abyss
Leviathan
What about ‘Heimat’ if you’ve got a few days to spare and can handle subtitles? It’s an extraordinary saga.
hee hee 
and let’s not forget 20,000 Leagues Under the Sea.
Now that’s deep!
What exactly is heavy about Happiness? Most American indie film-makers suffer from the notion that off-beat ideas involving sex and violence* are per se interesting and profound. I see it as largely reactionary film-making as they are willing to talk about the stuff Hollywood shies away from. But, what is more important is the content than the intention. Happiness barely touches upon the notion of happiness and doesn’t discuss the philosophy of happiness in any meaningful way. But, it has a guy starved for sex (I think there is the obligatory masturbation scene), and a father who is a pedophile. And, that is a profound film on the nature of happiness? Throw in the stilted acting and forced plot and it makes one hell of an idea gone awry.
- Crash was a film dealing only with sex and violence that could pull it off.
And, while Mystery Train was an interesting film, it is more like a collection of short stories than a philosophical novel.
Ju Dou is also primarily a tale. If one were to list films, which primarily told stories, only because there were very well-made, this thread will be useless. I grant that all well-made films make you think, but I think we ought to stay away from those which do not necessarily focus on the issues of purpose and nature of life, existence, death etc.
I tend to agree with this as a general principle. A movie should be engaging (and preferably entertaining) on the surface, and possibly imply deeper or more thought-provoking elements.
If you can tolerate dubs or subs, I would recommend a couple of anime films:
Spirited Away, which on the surface is a faerie tale about a modern child who strays into the realm of spirits, deals with themes of identity, loss of identity, and alienation from a number of very interesting angles.
Ghost in the Shell uses a cyberpunk cop story to deal with fundamental questions about the nature of life and of humanity. The philosophical elements are discussed openly a bit more than I generally prefer, but that’s acceptable because they are fairly important to the plot and the characterization.
Metropolis (both the original and the anime)
1984
Brazil
Clockwork Orange
Dead Man
Blue Velvet
You might be better off just reading the book for these two. Brazil is a good choice though. A magnificently inventive film, I highly recommend it. Try to watch the directors cut if you can.
As stated earlier if you liked the Matrix, try Dark City. A lot of the same themes of the Matrix, but IMHO, a deeper more satisfying film.
If you want some films that are really heavily into philosophy I recommend:
My Dinner With Andre an amazingly original film about two old friends meeting to have dinner for the first time in years. A unique film that manages to make the concept of the two opposing philosophies work.
Or
Waking Life A wonderfully inventive film that takes you into the dream of its protagonist. Very cool, very fun, and completely philosophical.
The first movie that popped into my head when I saw this thread was Fearless with Jeff Bridges.
Try The Rapture. It is a great, very underrated movie starring David Duchovny and Mimi Rogers. I hate to say too much about it for fear of giving things away, but suffice to say it will leave you thinking for a long time.
I second Dead Man, Jim Jarmusch’s best movie and so deep you’ll have to rewind it and watch it again right away, even though the though horrifies you.
Jacob’s Ladder is also very interesting and operates on many levels. Also, The Usual Suspects is a movie that has to be watched at least twice, but will yield surprises and consternation after many viewings.
I liked Jabba’s point - most good films are potentially deep, depending on the viewer. Some of the ones that set out to be profound, like Waking Life, end up going in circles (IMHO) (or maybe it was the jiggling - I just hated that film!).
However I also like your question - here are some of my favorites. Some of them hit it head-on, others are more oblique (I think I put these in order from dense to light).
My Dinner With Andre
Being There
Joe Gould’s Secret
Koyaanisqatsi
The Apostle
The Mission
Ordinary People
Amadeus
Glengarry Glen Ross
Jacob’s Ladder
Chariots of Fire
The Purple Rose of Cairo
Bullets over Broadway
Slacker
Harold and Maude
Big Night
Lost in America
Defending Your Life
Down and Out in Beverly Hills
Blade Runner
Total Recall
Read alongs - Tao Te Ching, all of J.D. Salinger, Illusions by Richard Bach. Also, Lonesome Dove by Larry McMurtry.
Check out Naked Lunch while you are at it. Also, Donnie Darko is very deep as is Memento. I’ll let you know some more if you want.
Mind Walk with Sam Waterston is like no other movie I’ve seen.
American Beauty has so many layers that I am overwhelmed everytime I watch it.
Wow, thanks for the many responses so far. I’ve got a lot of watching cut out for me; hopefully it will be a long summer.
I should note that I am not simply looking for a film that makes me think in order to understand it, but rather one that makes me think about something philosophical. So, I’m not looking for mysteries or twist endings really. It doesn’t even necessarily have to be complicated or subtle, as long as it raises questions in an interesting way. I imagine, though, that the films that do this best will be complicated and subtle.
Jabba: I don’t quite follow what you mean about fiction not being the best way of transmitting ideas. What if I want to learn about the philosophy of (say) Camus? I understand what you mean about going to the original source; maybe if I find something in a film that I like, I can ask where I can find out more about it.
SwimmingwithChickens, you’re spot on about my liking Dark City. It’s more for its imagery and pacing than its philosophy (which I thought it kind of missed), and it’s a fascinating film.
I went to the video store to get Crimes and Misdemeanors but it was checked out. The first film mentioned in this thread that they had was Raging Bull, so I’m going to start with that.
For something thought-provoking (which is a little different than “deep”), try these two:
Mondo Cane, which shows shocking images from, say, Africa or New Guinea and then juxtaposes them with comparable images from Europe or America and dares the viewer to say why one is more shocking than the other.
Dr. Death by documentarian Erol Morris. From an interview with the director, here’s a (hazily remembered) quote: “in Schindler’s List, Steven Spielberg had what I think is a very interesting idea, that anyone can be a hero. In Dr. Death, I have what I think is an even more interesting idea, that anybody can think he’s a hero.” A sad, unflinching look at a man who designs electric chairs for a living and fancies himself to be some kind of Holocaust debunker. Self-deluded losers are Morris’s stock-in-trade.
Then I reiterate my suggestion (seconded above by someone else) of Mindwalk. And I strongly second the suggestion for Waking Life.