You might want to check out Amelie or Royal Tenenbaums if you haven't seen these films yet, or Three Kings. Dark City and Solaris make good companion pieces to Blade Runner and 2001 if you like those films. Also, if you enjoyed Crouching Tiger Hidden Dragon then Hero makes an excellent follow-up, or for a more action-oriented version House of Flying Daggers. For character dramas of the highest quality check out anything by PT Anderson--Boogie Nights and his severly underrated debut, Hard Eight, are great but his best film is Punch Drunk Love IMO. And this is a star wars site, but i am always amazed at how few Star Wars fans have seen Lucas' previous masterpieces, THX 1138 and American Graffiti--if you haven't then do yourself a favor and check these out. Also while on the subject of Star Wars, if you haven't you might want to see the film that robbed it of its Oscar in 1977--Annie Hall.
Bio-pic wise i always have found Pollack to be very compelling. its like Raging Bull--with paint!

Theres also John Cusak's very interesting film Max in which he plays Hitler as a young art student in Berlin. And while on the subject of art bio-pics i guess i should mention Girl with the Pearl Earring as a great yet quiet film. Frida makes a more eccentric film if you want something a bit more lively.

For weirder stuff i recommend Pi (well, its spelt with the mathematic symbol), Being John Malcovich, or Evil Dead 2 (a little gory but its ultimately a slapstick comedy); Tetsuo is good too if you can stomach it, and Bergman's Persona still impresses me with its abstractness.

Inconvenient Truth is also an excellent film that is sure to create discussion afterwards. If you like documentarys i also highly recommend Fog of War, and the recently released War Tapes is pretty interesting too.