I'll try to give you some that I think no one else will recommend, as I'm sure most, if not all, are going to say Neon Genesis Evangelion, Death Note, Code Geass, ect.
Berserk - Berserk is a very mature anime, and an more mature manga. The story follows Guts as he travels with a band of mercenaries called the Band of the Hawk. It is not an anime for the faint of heart.
Serial Experiments Lain - SEL may not be as mature as say Berserk is for some people, but the ideals that are played out in the show are on a very mentally deep level. Within a minute into the first episode there is a suicide, to give you an idea.
Blue Gender - Blue Gender is a curious anime that doesn't truly involve any immoral discussions, IMO. It is a story of Earth losing control of its humanity against n invading alien force, and how the humans plan to retake Earth.
Type/Moon's Tsukihime, Lunar Legend and Fate/Stay Night - Both of these titles are really good anime, coming from Type/Moon's Video Game series. Tsukihime, Lunar Legend follows the story of Shiki as he travels around with an Elder Vampire in the hunt of a rouge killer who is suspected to be a Vampire. Fate/Stay Night follows Shirou and Saber as they try to win the Holy Grail War.