Using complex description of div and curl in 2d?

[Mappe]

In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z)), where f(z,z) is just f(x,y) expressed in z and z conjugate (z). Is there any way of proving the fundamental properties of div and curl and/or understanding them better by looking at d/dz f(z,z)?

 

[Mappe]

some characters did not write out, f(z,z_) its supposed to say, with z_ being complex conjugate