A generalized Dirac operator

Let $$(M^4,g)$$ be a closed four-dimensional Riemannian manifold and $$J$$ be an almost complex structure on $$M$$. Then for normal coordinate $$e_1,\dots e_4$$ at a point $$m,$$ and for a section $$\alpha$$ of a $$\mathrm{Spin}^\mathrm{c}$$ bundle $$V$$ on $$M$$, one can define the operator: $$\begin{equation*} D_J(\alpha)=\sum_i J(e_i)\cdot \nabla_i\alpha \end{equation*}$$

This can be seen as: $$\begin{equation*} \Gamma(V)\xrightarrow{\nabla}\Omega^1(M)\otimes\Gamma(V)\xrightarrow{J}\Omega^1(M)\otimes\Gamma(V)\xrightarrow{\text{Cliff. mult.}}\Gamma(V) \end{equation*}$$

Does this operator appear in the literature? If $$D$$ denotes the usual Dirac operator and $$D\alpha$$ vanishes, can we say something about $$D_J\alpha?$$

Comment: At a first glance this reminded me about the twisted differential, one defines in Kähler geometry: $$\begin{equation*} d^c:=\sum_i Je_i\wedge \nabla_i \end{equation*}$$ I am not sure if this helps in any way though.