Freya Holmer, of The Continuity of Splines [0] fame, recently gave a great conference talk about how quaternions naturally arise from the desire to define the multiplication of two vectors if you just agree to one new axiom: v^2 = ||v||^2, or "the square of a vector is equal to the square of its norm". It was fascinating and entertaining and I highly recommend anyone who finds this stuff interesting but confusing (like I do) to check it out: https://www.youtube.com/watch?v=htYh-Tq7ZBI