In the original comics and in the newly released photos of Benedict Cumberbatch from the movie version, Dr. Strange wears an outfit with an excessively high collar. Does this high collar have a purpose and, if so, what is it?
The collar, along with the cape attached to it, is actually an arcane artifact - one that responds to the mind of its wearer - called The Cloak of Levitation. The high collar itself does not serve a purpose, but that is indeed how the cloak was originally designed when it first appeared in Strange Tales #127 (December 1963). Even before the cape's debut, Strange wore a tunic which also had a high collar:
Speculation: At the time, high or "popped" collars were seen as a symbol of rebellion and "coolness" in American society. Doctor Strange has always been somewhat of an outsider in the Marvel universe, so perhaps that is what the cloak's collar was meant to convey.
Update: As an added bonus, there is a reference to this in the film itself...
At the moment where Strange decides to give up his old life and embrace his new role as a sorceror, he is staring into a mirror with the collar laying nearly flat. He pops the collar up in a "Let's do this" manner before proceeding to go kick magical butt.