Suppose we have a material which has a very, very high refractive index, say 30-50. The critical angle for this material would be very small (1.5-2 degrees). Will the said object be even visible at all, given that most of the light that enters it would not be able to come out again, at least until a few thousand reflections? Can anyone imagine/describe what it would actually look like?
What makes diamond so attractive (at least to some who have bought into the marketing craze created by the de Beers cartel) is not just the refractive index but the dispersion (aka "fire"), which is among the highest of all gemstones. It's also important to the quality of a diamond to be as translucent as possible, especially for the larger stones and the cut is of extreme importance. A poorly cut stone with the wrong facet angles does not produce the sparkle that is considered a necessary quality criterion. In other words: a jewelry diamond is more than just its material properties, it's also an optimized optical system.
As a consequence for your hypothetical material it would be important to have extremely low absorption, otherwise there would be little if any light coming out. I don't know how dispersion would play out and what the ideal "cut" would have to look like. If anything that sounds like a non-trivial optics problem.