Girls with purple hair are the Ojou or Shrinking Violet Kuuderes
I'm making this a more general trope, like Rose-Haired Girl. Still, this isn't a dumping ground for purple haired characters, they have to fit one of the descriptions.
Animation uses impossible natural hair colors to help differentiate characters who who may otherwise look quite similar. Some of these hair colors are indicative of certain things about those characters. Purple hair is no different and is the intersection of two opposite hair-color tropes: Shy Blue-Haired Girl and Rose-Haired Girl. Violet-haired girls can favor either the passion of a Fiery Redhead, or the manner of a cool Ice Queen. Purple Haired Girls can be associated with royalty and all the things that come with it, including wealth, status, sophistication and sometimes arrogance. With this association they may be princesses, or act as if others should treat them as such. Being Posh, Vain, Snobbish or Superior are also common. On the other hand however, they can be Elegant, Modest, and Kind. It's essentially the difference between a Royal Brat and Princess Classic. It's not uncommon for such characters to have a Noblewoman's Laugh, or speak in a very formal manner. A Hime Cut, is also very common for such characters, the hair-cut itself one associated with nobility and royalty in Japan. Shrinking Violet, this is such a well known phrase in the west that often purple-haired characters in Western Animation will be meant to allude to it. In this case the characters may be a quiet Kuudere or Emotionless Girl, or have No Social Skills, fearful of interaction with other people. Peek-a-Bangs are a common associated hair style, as well as speaking in a quiet voice, or not speaking much at all. common Mary Sue trait as well. Why? Because it's cool and unusual, probably. Purple hair is also strongly related to Tall, Dark and Bishoujo, because Purple Is The New Black. As black is very difficult to render on screen, characters whose hair would be understood as black hair if it were in Real Life, may be colored purple. Likewise if a character has black hair in another media, when it is being adapted to animation it's hair may suddenly be purple. A Subtrope of You Gotta Have Blue Hair. Compare Shy Blue-Haired Girl and Princesses Prefer Pink. Contrast Rose-Haired Girl. Has nothing to do with Elderly Purple Haired Ladies.
Examples:Anime And Manga