The Waltz Disney of the east, Hayao Miyazaki has for too long been ignored by western eyes. A man who has personally drawn and supervised all of his feature films ( a claim Waltz Disney could never adhere to), is surely the greatest Anime director/director period of all time. His exuberant take on life and his odd penchant for Lynch-like fatalism makes him a perfect candidate for scenester adoration. The pimp daddy has produced many cult classics, from My Neighbour Totoro to Princess Monoke( google now). But a must watch is "Spirited Away".
Commonly quoted as one the greatest anime flicks ever, the use of colour, traditional Japanese mysticism and good old fashioned good versus evil, makes this movie a must see. Although if you want a movie your less alien to, try "Howls Moving Castle", a western novel based on humorous wizardly, it entails an all star cast of Christian Bale, Jean Simmons and Emily Mortimer. (Emily's father is an fascinating character, read his books!)
Basically, if you don't know, get to know.