SUMMARY Modelling marine controlled-source electromagnetic (CSEM) responses in the fictitious time domain is a novel approach, which facilitates the full exploration of EM diffusive properties in the fictitious wave domain (FWD). Concepts, such as reflections, refractions, diffractions and transmissions, which are used for the analysis of elastic wave propagation can thus be adopted in FWD for interpreting CSEM data. In this paper, we use a high-order finite difference time domain (FDTD) algorithm for modelling marine CSEM responses in both the fictitious time domain and the diffusive frequency domain. A complex frequency shifted perfectly matched layer (CFS–PML) boundary condition is adopted to the FDTD modelling. We demonstrate the performance of the CFS–PML boundary condition and validate the high-order FDTD code in the FWD with the half-space sea water model and in the frequency domain with the 1-D canonical reservoir model. We investigate and analyse the propagation characteristics of electromagnetic fields in the FWD, where we apply wave propagation concepts to interpret marine CSEM data. Similarities between wave and field propagations relevant for marine CSEM data are demonstrated through several 1-D to 3-D numerical examples.