In mathematical physics, non self-adjoint operators often arise in connection with processes that do not conserve energy. From the mathematical point of view, such operators are of interest because they arise as the quantizations of complex-valued symbols, and the associated classical dynamics must be extended into the complex domain. In this talk, I will discuss the special class of non self-adjoint pseudodifferential operators with double characteristics, and I will present some new results on L^p-bounds for eigenfunctions of such operators in the semiclassical limit. The main tools used are the Fourier-Bros-Iagolnitzer (FBI) transform and microlocal analysis in exponentially weighted spaces of holomorphic functions.