Möbius Transformations Revealed is a wonderful, short film that illustrates the beautiful correspondence between Möbius transformations and motions of the sphere.
The film, created by mathematicians Douglas N. Arnold and Jonathon Rogness, has been viewed over 1.5 million times on YouTube, and is among one of the YouTube top favorites of all time.
A high resolution version along with other information can be found here.
If you are interested in finding out more, there is no better place to start than Tristan Needham’s amazing Visual Complex Analysis.