Divergence, curl, and gradient of a complex function From an answer here I got 
Green's theorem for functions in the complex plane 
$$
\oint f(z) \, dz = 
i \iint \left( \nabla f \right) \, dx \, dy = 
i \iint \left( 1 {\partial f \over \partial x} + 
i {\partial f \over \partial y} \right) \, dx \, dy
$$
Which leads to the well known Cauchy's integral theorem
$$
\oint f(z) \, dz = 
\iint 
\left( \frac{- \partial f_x}{\partial y} + \frac{- \partial f_y}{\partial x}  \right)+
i \left( \frac{\partial f_x}{\partial x} + \frac{- \partial f_y}{\partial y} \right)  \, dx \, dy
$$
From which I then get
$$
\oint f(z) \, dz = 
\iint \left( 
\nabla \times f +
i \nabla \cdot f 
\right) \, dx \, dy 
$$
 I'm hoping someone here can tell me whether I'm on the right track or not.
Keep in mind that
$$\nabla =  1 {\partial \over \partial x} + 
i {\partial \over \partial y}$$

I'm not a physicist, but I think that gradient, curl, and divergence are strictly for a real $d$-dimensional environment, in particular for $d=2$ and $d=3$. I have never met your strange complex definition of $\nabla$.
On the other hand it is of course possible to prove the Cauchy integral formula using Green's theorem in the form
$$\int_{\partial \Omega}\bigl(P(x,y)\>dx+Q(x,y)\>dy\bigr)=\int_\Omega(Q_x-P_y)\>{\rm d}(x,y)\ .\tag{1}$$
Write your analytic $f$ in the form $f=u+ iv$ as well as $dz$ in the form $dz=dx+i dy$. Then by definition of complex line integrals you have
$$\int_{\partial\Omega}f(z)\>dz=\int_{\partial\Omega}(u\>dx-v\>dy)+i\int_{\partial\Omega}(v\>dx+ u\>dy)\ ,$$
to which you can apply $(1)$ separately. Finally the CR equations will come to your rescue.
