Sophisticated anime. iin?

Evangelion - A post-apocalyptic/psychological horror/scifi about kids piloting giant robots.

Higurashi no Naku Koro Ni - Kids brutally murdering each other out of paranoia. Involves time traveling shenanigans.

Elfen Lied - Oppressed mutant humanoids brutally murdering normal people with their powers.

Puella Magi Madoka Magica - It's basically a Horror/tragedy version of "magical girl" shows like Sailor Moon for example.

Shadow Star NaruTaru (Absolutely NOT to be confused with the more mainstream and infinitely more childish Naruto) - it's basically a horror/tragedy version of Pokemon.

Bokurano - From the same creator as Shadow Star, Bokurano is similar to Evangelion in that it's a horror/tragedy involving kids piloting giant robots.

Perfect blue - Mima Kirigoe, a member of a pop-idol group called "CHAM!", decides to quit singing to pursue a career as an actress. Some of her fans are displeased with this decision though, particularly a stalker named Me-Mania. As Mima progresses into her new career, those close to her are violently murdered one by one, and Mima begins to lose the ability to distinguish reality from fantasy.

Akira - It's the year 2019, thirty-one years have passed since the start of World War III. A top-secret child with amazing powers of the mind breaks free from custody and accidentally gets a motorcycle gang involved in the project. This incident triggers psychic powers within one of the gang members, Tetsuo, and he is taken by the army to be experimented on. His mind has been altered and is now on the path of war, seeking revenge on the society that once called him weak.

Beluga - A horror spin on the classic tale of "The Little Match Girl".

Paprika - In the near future, a revolutionary new psychotherapy treatment called PT has been invented. Through a device called the "DC Mini" it is able to act as a "dream detective" to enter into people's dreams and explore their unconscious thoughts. Before the government can pass a bill authorizing the use of such advanced psychiatric technology, one of the prototypes is stolen, sending the research facility into an uproar. In the wrong hands, the potential misuse of the device could be devastating, allowing the user to completely annihilate a dreamer's personality while they are asleep. Renowned scientist, Dr. Atsuko Chiba, enters the dream world under her exotic alter-ego, code name "PAPRIKA," in an attempt to discover who is behind the plot to undermine the new invention.

Yukidoki - The short film tells the story of a young boy, who as a result of seeing a dog's corpse, becomes interested in the bodies of living things and starts to develop phobias. Before long all manner of things begin to appear grotesque to him.

Judging by the preferences you've listed, I have a few anime in mind you might enjoy.

Higurashi no Naku Koro Ni (and Kai): This one was already listed by someone else, but I have to recommend it. It's easily the most well known horror anime out there.

Shiki: This one's a vampire story; but don't worry, it's the complete opposite of Twilight. Basically, a family of vampires moves into a rural village and all hell breaks loose.

Princess Tutu: Now I know what you'll think as soon as you read the title. But don't. The show starts out innocently enough, but as it starts going, things take a dark turn. A great storyteller died while in the midst of writing the book "The Prince and the Raven". Because the story was never finished, inside of the book the prince and raven fight endlessly. Finally growing tired of the endless battle, they escape from the pages of the story and into the real world. The prince ultimately sacrifices himself by shattering his own heart to defeat the Raven. And that's just where the story begins. The story is essentially a tragic fairy tale, and it definitely works from that angle, but it's also simple enough to be timeless. Also, the soundtrack is amazing, comprised of classical music that gives every scene incredible power.

definitely go for One piece if you love adventure and would like to see a different world which is amazing , like how do creators think of this. Its all planned in their mind, its so engrossing. Naruto as well. They sure are stories i love that aren't perverted, their the funny kind with a plot so intriguing.

Hentai is a pretty good anime.

Tokyo Ghoul, Zankyou no Terror

Future Diary, Death Note, Gurren Lagann (the best anime ever), Sword Art Online (sort of), Outlaw Star, Blue Gender, Cowboy Bebop, Solty Rei, Eureka Seven. I hear Laughing Under the Clouds is one, but I haven't watched it yet.

Hmm the one that first came to mind i almost find hard to recommend neon genisis evangellion, its so flawed, so unbelievably flawed, the early parts of the season are deliberately misleading with very subtle foreshadowing I didn't pick up on until the second go, some of the content that I later found out was intentional really made me want to stop watching, there's way too much fan service, some themes are mishandled and the final two episodes are a result of the creators mental breakdown and an entire film was made by him after to actually clarify what the fuck happened, if it wasn't for my brother telling me that the series changed drastically halfway through I would have stopped watching

But it's a masterpiece, it's incredibly intelligent and deals with some heavy subject matter that became clearer the more I thought about it, so definitely watch it, but maybe after you've familiarised yourself with some anime conventions, its one to come back to

My recommendation for first anime would be the film grave of the fireflies, its probably the most conventional and westernised anime I've ever seen, it'd melancholic and tradgic yet at the same time strangely beautiful, if you want something more lighthearted go with my neibour tototro, its the most charming family film I've ever seen but with some darker subtext,

The original ghost in the shell film is a another good starting point and if you enjoyed that maybe the TV series, ghost in the shell stand alone complex

Paprika
Millennium Actress
The Wind Rises
Grave of Fireflies

Not just good anime, but top tier films.