Bicomplex Numbers Representing 3d Rotations

Question

One of the primary applications of quaternions is the calculation of rotations in $\mathbb{R}^3$; they can be intuitively realized on four-dimensional spherical coordinates $(r, \phi_1, \phi_2, \phi_3)$. A bicomplex number $(w, z)$ where $w, z \in \mathbb{C}$, can likewise be shown to reside on two polar planes $(r_1, \theta_1)$ and $(r_2, \theta_2)$. (The argument itself will then be a complex number as proved in Bicomplex Numbers and Their Elementary Functions.) Is there an existing method to represent three-dimensional rotations with bicomplex numbers?