We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schrödinger equation to the magnetic Schrödinger equation. That is in presence of a magnetic potential. We establish observability inequalities and exponential stabilization by extending the usual multiplier method, under the same geometric condition to that needed for the Schrödinger equation. We also prove, with the help of elliptic Carleman inequalities, logarithmic stabilization results through a resol-vent estimate. Although the approach is classical, these results on logarithmic stabilization seem to be new even for the Schrödinger equation.