We present a stochastic method for solving the time-dependent Schrödinger equation, generalizing a ground state full configuration interaction quantum Monte Carlo method. By performing the time integration in the complex plane close to the real-time axis, the numerical effort is kept manageable and the analytic continuation to real frequencies is efficient. This allows us to perform ab initio calculation of electron spectra for strongly correlated systems. The method can be used as a cluster solver for embedding schemes.