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.