What is the correct generalization of the Wirtinger derivatives to arbitrary Clifford algebras?

Question

In the complex numbers setting, the two Wirtinger derivatives are defined as:

$\frac{\partial}{\partial z}= \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) \quad \quad \frac{\partial}{\partial \bar{z}}= \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right)$

(see for reference the related question on Mathoverflow).

What is the correct generalization of these derivatives in a Clifford Algebra $Cl(p, q)$?

On the one hand, any such algebra has an involution which is the exact analogue of complex conjugation,so it would appear as if the generalization is straightforward.

On the other hand, I have the lingering suspicion that there is more to it, due to the fact that, unlike some special case such as the quaternions, the standard basis of the Clifford algebra has several $e_i$ that square to $+1$ and several others which square to $-1$.

This trivial observation leads me to think that perhaps there are many Wirtinger derivatives, not just two.

NOTE: The motivation behind this question is my previous one on Feuter regularity vs differentiability: as the standard proof in $\mathbb{C}$ uses the Wirtinger derivatives, I think that a clarification here may bring me closer to shed some light on the other question as well,

Answer 1

I guess there is more than one generalization of the two Wirtinger derivatives, but the most important one for me is the following (where I restrict to the positive definite case): In even dimension the spin representation splits into two irreducible representations $S^+\oplus S^-,$ and the (raw) Dirac operator acts as $$\mathcal D^\pm\colon \Gamma(.;S^\pm)\to \Gamma(.;S^\mp).$$ In the case of a Riemann surface equipped with a compatible Riemannian metric, these two operators are naturally given on the bundles $S^+=S$ (a spin bundle, i.e. $S^2=K$,) and $S^-=\bar S\cong S^*$ as follows: $$\mathcal D^+=\bar\partial\colon \Gamma(.;S)\to\Gamma(.;S\bar K)=\Gamma(.;\bar S)$$ and $$\mathcal D^-=\partial\colon \Gamma(.;\bar S)\to\Gamma(.;\bar S K)=\Gamma(.; S),$$ i.e. they are somehow the natural Wirtinger derivatives.

You can find more details on this in the book "Dirac operators in Riemannian Geometry" by Thomas Friedrich, for example.