I am trying to understand more about quaternions and I was watching the series of videos from Norman Wildberger, in particular I am a bit stuck on this video: https://www.youtube.com/watch?v=uRKZnFAR7yw
You can see he associates to a complex number a rotation $\varphi$. I forced myself to do some test with numbers, but assuming I have whatever complex number (i.e. $z=3+2i$) and I want to rotate it about the origin of $45^\circ$ what I tried to do is to use the parametrization of the unit circle displayed in the screenshot and enforced the $b/a = 2t/(1-t^2) = 1$ (the slope/trigonometric tangent) to find the complex number that would make such rotation, but that is going nowhere for me.
Probably I misunderstood what he is doing there but I still not see how I can use all of this theory in practice.