I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true division algebra in $\mathbb{R}^3$. Likewise, Liouville's theorem requires all three-dimensional conformal maps to be a composition of Mobius transforms. If I sacrificed commutativity, I could work with the Quaternions, which are a true division algebra; however, bicomplex numbers in $\mathbb{C}_2$ preserve commutativity at the expense of allowing zero divisors. I found a helpful diagram comparing the algebras at the following post.

A Comparison of Algebras Diagram

I am particularly interested in producing a biholomorphic or holomorphic mapping from a subset in one of these algebras to a subset of $\mathbb{C}$ using contour integration. I feel that the following paper would be more suited for defining the Cauchy-Riemann equations in the quaternions as opposed to the multicomplex numbers. I assume that helpful properties such as infinite differentiability as a result of the generalized Cauchy integral formula will not hold, especially in the case of multicomplex numbers when the projection of the contour onto one complex plane yields intervals of overlap or self-intersection.

Could the bicomplex numbers be trivially restricted to represent polar coordinates ($r$, $\theta$, $\varphi$) in $\mathbb{R}^3$. I only ask about this seemingly contradictory result as the Mandelbulb was defined using a three-dimensional analog of complex multiplication.

Furthermore, if the notion of contour integration were to exist in either of the quaternions or bicomplex numbers, would Stoke's theorem for 1-manifolds be applicable? Could a contour in this algebra even be shown to be diffeomorphic to an interval residing in $\mathbb{R}$?

I intend to avoid cross-posting, and for those interested, I will direct you all to my similar question posed on StackExchange mathematics.

Thank you all.

I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true division algebra in $\mathbb{R}^3$. Likewise, Liouville's theorem requires all three-dimensional conformal maps to be a composition of Mobius transforms. If I sacrificed commutativity, I could work with the Quaternions, which are a true division algebra; however, bicomplex numbers in $\mathbb{C}_2$ preserve commutativity at the expense of allowing zero divisors. I found a helpful diagram comparing the algebras at the following post.

I am particularly interested in producing a biholomorphic or holomorphic mapping from a subset in one of these algebras to a subset of $\mathbb{C}$ using contour integration. I feel that the following paper would be more suited for defining the Cauchy-Riemann equations in the quaternions as opposed to the multicomplex numbers. I assume that helpful properties such as infinite differentiability as a result of the generalized Cauchy integral formula will not hold, especially in the case of multicomplex numbers when the projection of the contour onto one complex plane yields intervals of overlap or self-intersection.

Could the bicomplex numbers be trivially restricted to represent polar coordinates ($r$, $\theta$, $\varphi$) in $\mathbb{R}^3$. I only ask about this seemingly contradictory result as the Mandelbulb was defined using a three-dimensional analog of complex multiplication.

Furthermore, if the notion of contour integration were to exist in either of the quaternions or bicomplex numbers, would Stoke's theorem for 1-manifolds be applicable? Could a contour in this algebra even be shown to be diffeomorphic to an interval residing in $\mathbb{R}$?

I intend to avoid cross-posting, and for those interested, I will direct you all to my similar question posed on StackExchange mathematics.

Thank you all.