I was first introduced to anime in the hospital 1 day after being born. I have devoted my life to the beautiful things that have been created. I, however am disappointed by the selection of anime my friends watch. The most popular amongst them is bleach, which repeats the same pattern in each episode. Or "Anime" that are no more than a cartoon being shown on a children's network. I have shown them what I watch and I've had a girl in tears watching my stuff. People need to branch out there anime! I am going to include a fair list of my favorites.(which none have any goofy art styles) and If these are something you don't normally watch then please compare them to what you Beowulf now.
Mushi-Shi (I will never find an anime like this in my lifetime)
Spriggan
Robotech chronicals
Armitage
Mad Lax
Vampire Hunter D
Ghost in the Shell
Noir
Cowboy Bebop
Black Lagoon
Elfen Lied
Sword of the Stranger
Grave of the fireflies (this is close, if not, my favorite anime of all time.)
ALL of hayao myazaki's work.
If you have any questions, or want. More recommendations ask.