In this paper, we present the stability analysis of the perfectly matched layer (PML) in two-space dimensional layered elastic media. Using normal mode analysis we prove that all interface wave modes present at a planar interface of bi-material elastic solids are dissipated by the PML. Our analysis builds upon the ideas presented in [SIAM Journal on Numerical Analysis 52 (2014) 2883-2904] and extends the stability results of boundary waves (such as Rayleigh waves) on a half-plane elastic solid to interface wave modes (such as Stoneley waves) transmitted into the PML at a planar interface separating two half-plane elastic solids. Numerical experiments in two-layer and multi-layer elastic solids corroborate the theoretical analysis, and generalise the results to complex elastic media. Numerical examples using the Marmousi model demonstrates the utility of the PML and our numerical method for seismological applications.