I understand that quaternions are sort of an extension of complex numbers in higher dimensions. If that's really the case conceptually (is it?), it must be possible to get back from the higher dimensional case to the lower one. How exactly?
Specifically, I have problems reconciling the 180 degrees rotation for inversion for complex numbers vs. 360 degrees rotation for inversion for quaternions. How does one generalize the 2D (complex numbers) case to the 4D (quaternions) case, or vice versa? I understand both these things individually well enough but have difficulties putting them together.