# Have complex manifolds with dual number structure on the holomorphic tangent bundle been studied?

If $$M$$ denotes a $$2n$$-dimensional real smooth manifold, then $$M$$ together with

• $$J\in\operatorname{End}(TM)$$ with $$J^2 =\mathrm{id}$$ (also called almost product structure on $$M$$) have been surveyed in detail in the rather dated Gadea, Cruceanu, Fortuny - A Survey on Paracomplex Geometry (1996)

and

• $$J\in\operatorname{End}(TM)$$ with $$J^2 =0$$ (also called almost dual number structure) have been studied in detail (within a larger class) by Shirokov, Shurygin, Vishnevskii and others.

Let $$M$$ now denote a $$4n$$-dimensional complex manifold and let $$J\in\operatorname{End}_{\mathbb{C}}(T^{1,0}M)$$ be an almost dual number structure on the holomorphic tangent bundle.

Have complex manifolds with such additional (almost-) dual number structures $$(M,J)$$ already been studied and if so, what are some references?

It was mentioned to me that (almost) dual number structures are important to physicists for some reason, but I am unsure if this was referring to the real or to the complex case or both. Also, he couldn't recall a precise reference, and my search has been unfruitful as I am quite unfamiliar with the mathematical physics venue.

Any pointers would be appreciated!