I will describe a certain generalisation of de Rham cohomology; things could be generalised further but I will stick to a concrete example. A $0$-double-form is a function on the complex plane $\mathbb{C}$. Using $z,\bar{z}$ as coordinates, the $1$-double-forms are spanned by $\eth z$ and $\eth \bar{z}$, where
$$ \eth z = \mathrm{d}z\otimes\mathrm{d}z,\qquad \eth \bar{z} = \mathrm{d}\bar z\otimes\mathrm{d}\bar z. $$
The $2$-double-forms are spanned by $$ \eth z\wedge \eth \bar{z}= \left(\mathrm{d}z\otimes\mathrm{d}z\right) \otimes \left(\mathrm{d}\bar z\otimes\mathrm{d}\bar z\right) - \left(\mathrm{d}\bar z\otimes\mathrm{d}\bar z\right) \otimes \left(\mathrm{d}z\otimes\mathrm{d}z\right). $$
The wedge operator is here defined to have all the expected properties. We can use the double exterior derivative $\eth$ to go from a $p$-double-form to a $(p+1)$-double-form. In particular, if $C$ is a $0$-double-form and $K=K_{zz}\eth z + K_{\bar{z}\bar{z}}\eth \bar{z}$ is a $1$-double-form, we define
$$ \eth C = \partial_z\partial_z C \eth z + \partial_{\bar{z}}\partial_{\bar{z}} C \eth\bar{z} $$ and $$ \eth K = (\partial_z\partial_z K_{\bar{z}\bar{z}} - \partial_{\bar{z}}\partial_{\bar{z}} K_{zz})\eth z \wedge \eth \bar{z}. $$ Clearly $\eth^2=0$. These notions can easily be extended to a higher dimensional space, or to triple-forms, quadruple-forms, and so on.
My question is about the general nature of this cohomology. Is it well studied, and if so what is it called? How well does Poincare's Lemma extend to this cohomology? Can we integrate double-forms? If so, what are the generalisations of things like Stokes' theorem?
Some background on why I am interested: gauge field theories are formulated in terms of forms, and it seems like general relativity might be able to be described in a similar way in terms of double-forms.
