  We establish the local well posedness of solution to the nonlinear
Schr\"{o}dinger equation associated to the twisted Laplacian on $\C^n$ in
certain first order Sobolev space. Our approach is based on Strichartz type
estimates, and is valid for a general class of nonlinearities including power
type. The case $n=1$ represents the magnetic Schr\"{o}dinger equation in the
plane with magnetic potential $A(z)=iz, z\in\C$.
