The IKKT matrix model has attracted considerable interest as a promising candidate for a non-perturbative formulation of superstring theory. In this model, we may study the scenario of the dynamical emergence of four-dimensional spacetime in superstring theory. Numerical simulations have been extensively used to study this nonperturbative effect on the Euclidean model. Recently, the Lorentzian model has also been studied, even though it has a strong complex action problem. This has been made feasible by using the Complex Langevin Method (CLM). In this talk, we provide an overview of this effort, and discuss the effect of adding a Lorentz invariant mass term to the original model.