How can one intuit complex numbers from quaternions?

Question

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.

Answer 1

Consider 3d space restricted to the $xy$ plane. Vectors on this plane are of the form $ui+vj$ in terms of quaternions, and they can be rotated by using quats of the form $\exp(k\theta/2)$. That is,

$$R(ui+vj) = e^{k\theta/2} (ui+vj) e^{-k\theta/2}$$

Take the latter exponential and see that $(ui+vj) e^{-k\theta/2} = e^{k\theta/2}(ui+vj)$. This can be verified, for example, by breaking $\exp(-k\theta/2)$ into sines and cosines, and using that $jk = -kj$ and $ik = -ki$. Hence, we see that the rotation takes the equivalent form

$$R(ui+vj) = e^{k\theta} (ui+vj)$$

which is, on its face, the rotation law for complex numbers. To complete the identification, multiply by $-i$ on the right:

$$R(u+kv) = e^{k\theta} (u+kv)$$

And we see here that $k$ on this plane performs the same role as the complex imaginary.

This approach makes clear that complex numbers and quaternions use the same rotation law, in a fundamental sense, but complex numbers have additional commutation properties that let them simplify down to the usual law (that doesn't involve half angles).

Answer 2

To get a complex number back from a quaternion $q= w + xi +yj +zk$ you just take the real part and any one of the imaginary components (i.e. $z = w + xi$). That subspace forms the complex numbers.

With respect to the rotation of a vector, it has to do with the interpretation of the geometry. One quaternion encodes an angle of rotation of $\theta/2$ but it is used twice in the computation (i.e. $p' = qpq^{-1}$). A complex number encodes an angle of rotation of $\theta$ and it is only used once in the rotation. So they really encode the same thing. We could have decided as a convention to encode a complex number for a rotation as $z = e^{i\theta/2}$. Then a rotation of $p$ would be $p' =zpz$, but since complex numbers commute, we don't have to do that. If we did choose that convention then the inversion of $z$ would be equivalent to a 360 deg rotation.