This is a simply brilliant bit of mathematical visualization. It you want a simple mental picture of Möbius transformations, this video will give it to you. Stunning.
The video is produced by Jonathan Rogness and Douglas N. Arnold. They have made available also for download a 130 Mbyte high res version, as well as the low-res clip on YouTube. There is a page for this video, which is produced under a generous Creative Commons license.
I picked this up from Good Math, Bad Math.