Given an integral of the form
\begin{equation}\int d\bar z\, dz\, \delta (\bar z \cdot A \cdot z-b)\,f(\bar z,z)\end{equation}
Where $z$ is a complex $n$ dimensional vector, and $A$ is an arbitrary, possibly complex matrix, how would you sort out the delta function so that it provides a constraint over each variable $z\_{i}$ and $\bar z\_{i}$ rather than a general constraint on the inner product? 
I guess it amounts to using a property analogous to $\delta(g(x))=\sum\_{i}\frac{\delta (x-x\_{i})}{|g'(x\_{i})|}$ but I'm still unsure how to formalize that. Any thoughts?