A non constant complex valued function with zero derivative I need an example of a non constant complex valued function whose derivative is zero at every point in in the complex plane.

A complex valued function on a connected set with derivative $0$ is a constant. (Refer Stein - Shakarachi Complex analysis Chapter 1 for proof.)
If the function is defined on disconnected regions then its derivative is zero need not imply function is constant.
For example take $f$ to be $1$ on $|z|<1$ and $2$ on $|z-3|<1$. Then $f$ is non-constant but $f'(z)=0$.
