I disagree, but then I used bivectors extensively in the geometric part of my mathematical research (back when) so I would. ;) y^2 = 1 is because it's a unit v

I disagree, but then I used bivectors extensively in the geometric part of my mathematical research (back when) so I would. ;) y^2 = 1 is because it's a unit vector. The fact that bivectors are the correct way to view rotations is evident if you look at higher dimensions. Bivectors work just the same way as described here when you want to work in R^n, you just add basis vectors beyond x, y, z. But the algebra they describe no longer forms an associative division algebra over the reals. There is no way to get to the algebra of the (generators of) rotations in higher dimensions from the quaternions. This is why i would say the quaternions (as built up from the complex numbers) are a non-sequitur and incidental to the structure of rotations. Then again, I would always argue that one should use complex numbers to think of rotations in R^2, so.... :)