Boy did this end up being a large project! I first came across quaternions in the context of math, as a noteworthy number system extending the complex number. Many programmers and animators may have come across them as the bit of black magic within 3d graphics engines that makes descriptions of rotations and orientations "just work".
In either case, there is a common frustration with not being able to think about four dimensions, and hence not being able to visualize what quaternion multiplication is doing. The aim of this project is to give a way to concretely think about this multiplication within the confines of three-dimensions. The first installment is a 30-minute video, the bulk of which builds up the idea of stereographically projecting a 4d hypersphere into the entirety of 3d space.
The next part gets into how quaternions apply to 3d rotation. This is introduced with a short video on YouTube, but things get much more interesting after that...
The bulk of this lesson is actually hosted off YouTube, where thanks to the fantastic web development wizardly of Ben Eater it is done in the form of "explorable videos" that you can interact with during the narration. To accommodate anyone landing on the page of explorable videos without having watched the ~36 minutes of YouTube content above, some of the explorables are a bit redundant with the previous video. But a little repetition never hurts when we're dealing with a difficult topic.