Molecular dynamics computer simulations have been conducted to examine the self-diffusion process for the liquid phase of the classical Dzugutov model. Mean-square particle displacements as a function of time have been evaluated over a wide temperature range, at reduced density 0.85, for both the continuous Newtonian trajectories and the corresponding piecewise-constant inherent-structure trajectories. Both representations yield the same self-diffusion constants but display distinct asymptotic offsets. These offsets possess different temperature dependences, with a crossover well into the supercooled liquid regime, close to reduced temperature T = 0.7. Lindemann ratios have been obtained for both the stable bee crystal and the liquid, showing a characteristic jump upon melting. Although its magnitude appears to be model-dependent, this jump signifies a marked difference in geometric character of the inherent-structure basins that respectively underlie the two phases, and that have correspondingly different interbasin transitions controlling the rate of self-diffusion.