  We study a stochastic Schr{\"o}dinger equation with a quadratic nonlinearity
and a space-time fractional perturbation, in space dimension less than 3. When
the Hurst index is large enough, we prove local well-posedness of the problem
using classical arguments. However, for a small Hurst index, even the
interpretation of the equation needs some care. In this case, a renormalization
procedure must come into the picture, leading to a Wick-type interpretation of
the model. Our fixed-point argument then involves some specific regularization
properties of the Schr{\"o}dinger group, which allows us to cope with the
strong irregularity of the solution.
