'polar' forms of quaternions

Question

We call a quaternion $q = q_0 + q_1 i + q_2 j +q_3 k$ purely imaginary if $q_0=0$. Here $q_0,q_1,q_2,q_3$ are real numbers, and $i, j, k$ the three imaginary units.

Is there a reference for the fact that any quaternion $q$ can be written as the product of a purely imaginary one $p$ times a phase $\psi\in (0,2\pi]$, $$q = p e^{i \psi}$$ or a discussion/proof available?

Answer 1

There is a $\psi\in [0,2\pi)$ such that $(q_0 + q_1 i)e^{-i \psi}$ is purely imaginary. Since $(q_2 j + q_3 k)e^{-i \psi}$ is of the form $rj+sk$, it follows that $p=qe^{-i \psi}$ is purely imaginary.