If you're looking for anime similar to Midori no Neko, you might like these titles.
Ryosuke is determined; he will find Bagi, and he will kill her. He has been searching for her for five years, but has known her much longer than that. Ryosuke had been a lonely child, as his parents were away working most of the time. He was given a kitten that he took quickly to, despite the fact that she didn’t look or act quite 'normal'. He named her Bagi, and they lived as best friends… until everyone around started to notice how different she really was and so she was discarded. Now, over a decade later, word has spread that Bagi has been terrorizing villages deep in the South American jungles and leaving many dead. With the help of a young village boy, Ryosuke sets out to stop her, but will he really have the heart to take her life?
Both anime were made in the 80's and were both directed by Osamu Tezuka. Also, both anime involve a man who is forced to confront his childhood friend (which in both cases is some sort of cat-like creature). If you like one, you will probably like the other.
Fudou Academy's Hajime Kindaichi is the grandson of the famous detective Kousuke Kindaichi. Though he often appears to be lazy in school and clumsy, he is actually a modern-day genius with an IQ of 180. Together with his childhood friend Miyuki, Hajime solves the most unsolvable murder mysteries with ease and grace. From missing students to wax museums, there's no case that Hajime Kindaichi can't solve!
Thirty years ago, Tokyo was destroyed in a massive earthquake known as The Great Fall. Today, the remains of the dead wasteland stand as the city of Kyuto, which is inhabited by few. In this city lives the head of the Hyuuga family, twelve-year-old Mayuki, who is able to understand the world around him with the help of extrasensory abilities. Along with Minori and the twins Kouta and Rakuta, Mayuki will solve the strange events taking place in the city; but who are the nefarious forces that are running the show?