Okay, let's see here...I'll give ya a list of things that you may enjoy along with a brief little synopsis.
Serial Experiments Lain - Probably my favorite anime. Some think it's brilliant (like myself), and others find it to be a bit bland. It's a great thought provoking series, set with a lot of technological ideas involving the internet, and some interesting questions about what makes someone really exist.
Boogiepop - Someone else mentioned this above. A fairly dark series with very good cinematography. The plot will keep your mind wondering, but overall it's not as complex as the likes of Lain.
Perfect Blue - One of the best suspense/thriller anime movies out there. Great plot that really keeps you wondering the whole time. However, this one is a bit on the graphic side, so if you're easily offended by sexual content it may not be for you.
Berserk - This is a very good action series, with a surprising amount of character development and a really twisted and erie plot. Lots of blood and violent fight scenes, and the end is pretty disturbing, so once again not for the squeemish (sp?).
Escaflowne - A very good mix of fantasy, action, and romance. I really can't say enough about this title...It has a great plot, wonderful character development, and an awsome soundtrack.
Magic Knight Rayearth - A happy and lighthearted fantasy series, but also serious. It's hard to say why this anime should be seen...nothing about it is extraordinary, but as a complete package well...it just makes you feel good inside hehe.
Kimagure Orange Road - Fairly serious romantic series, but it does have it's fun moments. I've not seen the TV series, but the OVA is very good, especially augmented by the KOR Movie.
Grave of the Fireflies - Extremely touching and well done movie set in WWII times. It's based around the lives of two children, and you feel very close to them by the end of the movie.
Oh My Goddess - 5 Episode OVA and a Movie. I love this series...it's a great mix of romance and comedy, fairly serious, you really care about the relationship of the two involved...As a bonus, the slapstick comedy is not nearly as prevelent as series like Love Hina.
Well...this is all I can think of at the current time, that you may like. Of course you can always find reviews and screenshots of anything before you buy it to get a better idea.
