We evaluate far-field tsunami hazard in the Indian Ocean Basin based on hydrodynamic simulations of ten case studies of possible mega earthquakes at the major seismic zones surrounding the basin. They represent worst-case scenarios of seismic rupture along the full extent of seismogenic faults having supported large earthquakes in the historical record. In a series of numerical experiments in which the source parameters of the 2004 Sumatra tsunami are allowed to vary one by one, while keeping the seismic moment and the fault orientation unchanged, we document that the main patterns of far-field tsunami amplitudes are remarkably robust with respect to nominal variations in such parameters as hypocentral depth, exact centroid location, and slip distribution on the fault plane. These results validate the concept of modelling case scenarios of potential future earthquakes whose source is by definition imprecise.We consider seismic sources located at the extremities of the 2004 Sumatra-Andaman rupture, namely along the southern coast of Sumatra and in the Andaman-Myanmar province; along the Makran coast of Pakistan and Iran; and also along the southern coast of Java, where the possibility of a large interplate thrust earthquake cannot be entirely dismissed. The results of our hydrodynamic simulations indicate that the distribution of maximum amplitudes in the Indian Ocean Basin is primarily controlled by the classical effect of source directivity, and additionally by refraction and focusing along bathymetric features. As a result, many provinces in the basin could be threatened by higher tsunami amplitudes than in 2004. This pattern is particularly important along the coast of East Africa, from Somalia to and including South Africa, in Madagascar and the Mascarene Islands, especially under a South Sumatra scenario involving an earthquake comparable to, or even possibly larger than, the 1833 event, whose epicentral area is widely believed to be under enhanced seismic risk as a result of stress transfer from the 2004 and 2005 ruptures to the northwest, possibly even in the wake of the 2007 Bengkulu earthquakes.