cv.complex variables – Bicomplex Conjugate Derivative

I have decided to first ask my question and second provide a list of steps I have already considered.

Thank you in advance.

Question: After reading the following paper on Bicomplex numbers, I am wondering if the derivative $frac{partial F}{partial Z^{dagger}}$ could be defined for a function $F: mathbb{BC} to mathbb{BC}$ as a means to generalize the notion of quasiconformality. Unfortunately, the fact that $bar{z}$ is not holomorphic appears to preclude the usage of the multivariate chain rule to calculate the derivative. I have written the four components of a bicomplex number as $x$, $y$, $k$, and $l$ as a 4-tuple in $mathbb{R}^4$. Could they be expressed in terms of $Z$ and $Z^{dagger}$ using an intermediate formula relying on some other notion than the complex conjugate? I wish that the dot product was defined for the underlying four tuples as one could trivially assert $left<1, 0, 0, 0right> cdot Z = x$.

Existing Thoughts: The notion of complex quasiconformality is defined as follows.

${mu}$ is a complex Lebesgue measure s.t. $sup{leftlbrace{mu}{left({x}right)}rightrbrace}<{1}$.

$$
displaystyle frac{{partial{f}}}{{partialoverline{{{z}}}}}={mu}{left({z}right)}frac{{partial{f}}}{{partial{z}}}nonumber
$$

${mu}$-quasiconformality may be ensured if $frac{{partial{f}}}{{partial{bar{z}}}}<frac{{partial{f}}}{{partial{z}}}$ in the strict sense. The existence of a measure ${mu}$ is then guaranteed as no statement of continuity is attested. The above is justified using the operator $frac{{partial}}{{partialoverline{{{z}}}}}=frac{{1}}{{2}}{left(frac{{partial}}{{partial{x}}}+{i}frac{{partial}}{{partial{y}}}right)}$, ${x}=frac{{{z}+overline{{{z}}}}}{{2}}$, and ${y}=frac{{{z}-overline{{{z}}}}}{{{2}{i}}}$. By the multivariate chain rule, we define $frac{{partial{f}}}{{partialoverline{{{z}}}}}$ as follows.

$$
displaystyle frac{{partial{f}}}{{partialoverline{{{z}}}}}=frac{{partial{f}}}{{partial{x}}}frac{{partial{x}}}{{partialoverline{{{z}}}}}+frac{{partial{f}}}{{partial{y}}}frac{{partial{y}}}{{partialoverline{{{z}}}}}=frac{{1}}{{2}}{left(frac{{partial{f}}}{{partial{x}}}+{i}frac{{partial{f}}}{{partial{y}}}right)}=frac{{1}}{{2}}{left({u}_{{x}}+{i}_{{x}}+{i}{u}_{{y}}-{v}_{{y}}right)}nonumber
$$

Note that this equation will equal $0$ provided that ${u}_{{x}}-{v}_{{y}}={0}$ and ${v}_{{x}}+{u}_{{y}}={0}$, which guarantee the Cauchy-Riemann equations.

Naturally, we ask if ${mu}$-quasiconformality can be generalized to a bicomplex Lebesgue measure ${mu}:{mathbb{{{B}}}}{mathbb{{{C}}}}to{mathbb{{{R}}}}$ with $sup{leftlbrace{mu}{left({Z}right)}rightrbrace}<{1}$ in the following sense. ${F}:{mathbb{{{B}}}}{mathbb{{{C}}}}to{mathbb{{{B}}}}{mathbb{{{C}}}}$ is a holomorphic function and $frac{{partial}}{{partial{Z}^{dagger}}}$ is defined by the same algebra as $frac{{partial}}{{partialoverline{{{z}}}}}$.

$$
displaystyle frac{{partial{F}}}{{partial{Z}^{dagger}}}={mu}{left({Z}right)}frac{{partial{F}}}{{partial{Z}}}nonumber
$$

We consider if such bicomplex quasiconformality guarantees complex quasiconformality on each of the complex components. A notion of a conformal mapping does not exist on ${mathbb{{{R}}}}$, so no analogue of this property exists on ${mathbb{{{C}}}}$.

Before proceeding, certain terminology will prove to be advantageous. ${F}{left({z}_{{1}},{z}_{{2}}right)}:{mathbb{{{B}}}}{mathbb{{{C}}}}to{mathbb{{{B}}}}{mathbb{{{C}}}}$ can also be written as ${F}{left({x},{y},{k},{l}right)}={left({U}+{i}{V}+{j}{K}+{i}{j}{L}right)}:{mathbb{{{R}}}}^{{{4}}}to{mathbb{BC}}$. Our notion of the components of a complex number from linear combinations of the conjugate generalizes to ${mathbb{{{B}}}}{mathbb{{{C}}}}$ as ${z}_{{{1}}}=frac{{{Z}^{dagger}+{Z}}}{{2}}$, ${z}_{{{2}}}=frac{{-{Z}^{dagger}+{Z}}}{{{2}{j}}}$, ${x}=frac{{{z}_{{{1}}}+overline{{{z}_{{{1}}}}}}}{{2}}$, ${y}=frac{{{z}_{{{1}}}-overline{{{z}_{{{1}}}}}}}{{{2}{i}}}$, ${k}=frac{{{z}_{{{2}}}+overline{{{z}_{{{2}}}}}}}{{2}}$, and ${l}=frac{{{z}_{{{2}}}-overline{{{z}_{{{2}}}}}}}{{{2}{i}}}$.

Where Things Break Down: The fact that the complex conjugate $overline{{{z}}}$ is nowhere differentiable presents a particular difficulty when attempting to apply the multivariate chain rule.

$$
displaystyle frac{{partial{F}}}{{partial{Z}^{dagger}}}=frac{{partial{F}}}{{partial{x}}}{left(frac{{partial{x}}}{{partial{z}_{{1}}}}frac{{partial{z}_{{1}}}}{{partial{Z}^{dagger}}}+frac{{partial{x}}}{{partialoverline{{{z}_{{1}}}}}}frac{{partialoverline{{{z}_{{1}}}}}}{{partial{Z}^{dagger}}}right)}+frac{{partial{F}}}{{partial{y}}}{left(frac{{partial{y}}}{{partial{z}_{{1}}}}frac{{partial{z}_{{1}}}}{{partial{Z}^{dagger}}}+frac{{partial{y}}}{{partialoverline{{{z}_{{1}}}}}}frac{{partialoverline{{{z}_{{1}}}}}}{{partial{Z}^{dagger}}}right)}+ \ frac{{partial{F}}}{{partial{k}}}{left(frac{{partial{k}}}{{partial{z}_{{2}}}}frac{{partial{z}_{{2}}}}{{partial{Z}^{dagger}}}+
frac{{partial{k}}}{{partialoverline{{{z}_{{2}}}}}}frac{{partialoverline{{{z}_{{2}}}}}}{{partial{Z}^{dagger}}}right)}+frac{{partial{F}}}{{partial{l}}}{left(frac{{partial{l}}}{{partial{z}_{{2}}}}frac{{partial{z}_{{2}}}}{{partial{Z}^{dagger}}}+frac{{partial{l}}}{{partialoverline{{{z}_{{2}}}}}}frac{{partialoverline{{{z}_{{2}}}}}}{{partial{Z}^{dagger}}}right)}
$$

For example, I cannot possibly see how to define the derivative of $bar{{z}_{{{1}}}}=overline{frac{{{Z}^{dagger}+{Z}}}{{2}}}$ considering that the complex conjugate is neither analytic nor holomorphic. (On $mathbb{BC}$, holomorphicity and analyticity are not logically equivalent.) Does anyone have an alternate idea to define quasiconformality for $mathbb{BC}$ in a meaningful fashion? Could the relationship to $bar{z}$ be avoided?