A $\mathbb{R^2}$ unit vector can be expressed as a complex number, using a $\textit{spiral phase quadrature}$ ( Larkin 2001 ): $$(x,y)\in\mathbb{R^2} \longrightarrow z \in \mathbb{C} =x+iy = e^{i\theta}$$ where $\theta$ has the same meaning than the $\theta'$ from polar coordinates.
Would it possible to express a $\mathbb{R^3}$ unit vector as a quaternion? $$(x,y,z)\in\mathbb{R^3} \longrightarrow q \in \mathbb{H} =ix + jy +kz = e^{j\theta}e^{k\phi}$$ where $\theta$ and $\phi$ are equivalent to $\theta'$ and $\phi'$ from spherical coordinates.
I know that an unit quaternion can be expressed as: $$q = e^{i\alpha/2}e^{k\beta/2}e^{j\gamma/2} $$ where $ \alpha$, $\beta$ and $\gamma$ are the Euler angles. but I would prefer to work in spherical coordinates. Is there an analogy to the spiral phase in 3D?
The traditional vectors are in quaternions represented in the following way for $(x,y,z)\in\Bbb R^3$, $xi+yj+zk\in\Bbb H$. However we have for all that $$e^{ix}=\cos x + i\sin x$$ with respective $i,j,k$. So if we try to get a product we run into the issue of that any product of $3$ such will still give a real number value, and if you try to only use $2$ of those basis then the last one will still appear except for a small set of fixed angles. As such you cannot do it as a mere product.