"Best" with regard to the criteria given would definitely have to be between the "Muslims" and "Alexander's Empire (Greece)" which I will respectively rename Ummayad and early Abbasid empires and the Macedonian/Hellenistic Kingdoms from Phillip (just as much if not more of a military genius than his kid) to the early successors of Alexander. The reason being is that these empires left a huge social and cultural footprint over large swatches of the world like that of Rome but did it much faster with much more explosive military success like the Mongols.

Not to say that the Romans didn't have fantastic military successes, but they would usually take a lot more time to consolidate their conquests and spread it out over a much longer period of time. Nor am I saying that the Mongols did not have any impact on the culture, because they were instrumental in joining the 'east' and 'west' as mentioned before, but they would quickly adopt the customs, language, and religion of the conquered.

Alexander's conquest of the old Persian empire absolutely made Hellenistic culture dominant in the Near East and it led to the primacy of the Greek language. In no way do I subscribe to the common thesis that the Hellenistic period was one of cultural stagnation compared to what came before, it outdid than the classic period of Greek city states! There's no doubt what influence Hellenistic culture had on the unconquered peoples of Rome and Carthage too.

The Arab conquests of the Near East and the Sassanid Empire is equally impressive, if not more, as they concurrently advanced on both fronts successfully. And then they went beyond into North Africa and Central Asia. The Battle of Talas River is probably the most underrated battle in terms of influence ever. Even Hellenization was not able to get rid of the Aramaic cognates, but after the Arab conquests, Syriac all but died out.

If I had to pick between one, I'd have to go with the Ummayad/early Abbasid empires as there is a lot more continuity between them, whereas immediately after Alexander's death, Egypt, the Near East and Persia, and Macedonia/Asia Minor would not be held by the same dynasty for any significant amount of time simultaneously. The Abbasids lost Spain immediately after removing the Ummayad dynasty, but would keep most everything else, and though they would see their empire fragment quickly in the next few hundred years, even then the newly independent Amirs would still recognize Baghdad for a bit longer, avoiding the interwarfare that plagued the successors.