After a lot of digging in google, I found this(eqn. 36, pg. 2849) paper, which gives a 'real' polar form of a quaternion. It states that, any quaternion q can be represented as $q=\vert{q}\vert{e^{i\phi'}e^{k\psi}e^{j\theta}}$ where phase $(\phi,\theta,\psi)$ is almost uniquely defined in the interval $[-\pi,\pi)\times[-\pi/2,\pi/2)\times[-\pi/4,\pi/4]$.
The formulas which they gave for getting those three phase angles were quite complex, and I'm wondering is there actually a way to convert a quaternion in polar form, back to rectangular form. If so, how?
Also, how to convert octonions to polar form?
