Elfen Lied (2004) - Season 1

The Diclonius, a mutated homo sapien that is said to be selected by God and will eventually become the destruction of mankind, possesses two horns in their heads, and has a "sixth sense" which gives it telekinetic abilities. Due to this dangerous power, they have been captured and isolated in laboratories by the government. Lucy, a young and psychotic Diclonius, manages to break free of her confines and brutally murder most of the guards in the laboratory, only to get shot in the head as she makes her escape. She survives and manages to drift along to a beach, where two teenagers named Kouta and Yuka discovers her. Having lost her memories, she was named after the only thing that she can now say, "Nyuu," and the two allow her to stay at Kouta's home. However, it appears that the evil "Lucy" is not dead just yet...