No. It’s more what the previous poster said about encoding rotation. It’s just not a xyz axes. It’s current, charge, flux as axes. The trig is how you collapse the 3d system into a 2d or 1d projection. You lose some information but it’s more useful from a spefic reference.
Without complex numbers you can’t properly represent the information.
Yeah. The thing that made me “get” quaternions was thinking about clocks. The hands move around in a 2d plane. You can represent the tips position with just x,y. However the axis that they rotate around is the z axis.
To do a n dimensional rotation you need a n+1 dimensional axis. So to do a 3D rotation you need a 4D axis. This is bassicly a quat.
You can use trig to get there in parts but it requires you to be careful to keep your planes distinct. If your planes get parallel you get gimbal lock. This never happens when working with quats.