Do you have any examples of piecewise-conformal homeomorphisms between polygons (possibly a reference in literature)? Thank you again for all of your insight.

Thank you for your prompt response! By piecewise linear homeomorphism from a finite polygon, do you mean a map from a polygon embedded in $\mathbb{C}$ to a polytope embedded in $\mathbb{R}^3$? Or, do you mean a map between polygons in $\mathbb{C}$ or $\mathbb{R}^3$? If I were to ask a follow-up question about purely conformal maps, would it be best practice to make a separate question?

Thank you, I reverted my question.

If you do not mind me asking, what differential geometry textbooks do you recommend? For a preliminary understanding, I have read M. Spivak's Calculus on Manifolds and J. Munkre's Analysis on Manifolds. Though K. Tapp's textbook Differential Geometry of Curves and Surfaces is quite recent, I found that his use of computer aided design for diagrams was particularly useful.

Yes, bicomplex holomorphicity is assumed. (I've edited my question.) To summarize my mistake, the operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline{z}}$ may still exist even if the complex derivative $\frac{df}{dz}$ is not defined for a given function $f: \mathbb{C} \to \mathbb{C}$. As a digression, a similar linear change of variables $x'=ax+by$ and $y'=bx-ay$ for $a, b \in \mathbb{R}$ is often used to solve first-order linear PDEs. Thank you for all of your help; in this fortunate turn of circumstance, quasiconformality generalizes quite nicely.

To expound on my original confusion, I knew that the derivative $f'(z)$ does not exist anywhere for the function $f(z) = \overline{z}$, so I wondered how one could evaluate $\frac{\partial \bar{z_1}}{\partial z_1}$. Thank you.

To answer your first question, I am looking for the latter: a definition of bicomplex quasiconformality that matches the existing definition of $\frac{\partial}{\partial Z^{\dagger}}$. Has such a method already been proposed in literature? Specifically, I was interested in other criteria (if any) for proving that bicomplex quasiconformality implies that each complex component is a quasiconformal map as well. I then wondered if bicomplex quasiconformality could be characterised in both a weak, global sense and a strong, component-wise sense.

Thank you for your insight. How did you calculate $\partial \bar{z}_1 / \partial Z^\dagger = 0$? I had thought that the result was undefined as I believed $\overline{Z^{\dagger} + Z}$ was not analytic after taking $2z_1 = Z^{\dagger} + Z \in \mathbb{C}$ for $Z \in \mathbb{BC}$. Could we treat $z_1$ and $\overline{z_1}$ as distinct variables? Originally, I had wanted to set all partials with respect to a conjugate equal to $0$ for simplicity; are you saying that such an assertion is valid after all? Thank you again.

Is there a general formula that could work for all polygonal geometries? Do you know of one?