Q. What was Miriam's and Aaron'c complaint against Moses in Numbers 11? Was it because their brother had married a non-Israelite Ethiopian or because he was a polygamist?
Tradition says that Moses married an Ethiopian (Cushite) woman from Meroë when he was Prince of Egypt. Quite possibly this was an arranged marriage, a part of the common practice in those days of the kings of subject peoples giving their daughters in marriage as part of the system of tribal alliances to prevent them going to war with one another. Or or it may have been a genuine marriage of love. One thing we do know is that the Hollywood version, The Ten Commandments, in which an Egyptian princess, the actual daughter of Pharaoh, tries to woo the handsome and dashing Moses, is simply a piece of American Protestant fantasy.
It is hard to believe that Miriam's and Aaron's opposition to Moses was on account of polygamy because polygamy was completely normative at that time and is the only way to account for the rapid growth of the Israelites in Egytptian captivity. More likely the objection was linked to the Ethiopian woman's connection to the old way of life they were fleeing, though it could have been racially motivated, since the Ethiopians were dark-skinned. Even the latter seems unlikely since we know that a good number of Egyptians joined with the Israelites in the exodus from their own country, doubtless impressed by the miracles they had witnessed and wanting to be on the 'right side'.
I give it only as an opinion, since we do not have enough data on the subject to arrive at any sure answer, but I suspect the Ethiopian was a princess accustomed to being treated royally and who expected, as a king's daughter, to have a position of authority in the camp of Israel as the leading prophet's wife. This might well have provoked resentment. Or I could be completely wrong. Perhaps the woman refused to become an Israelite by adoption into the covenant? Or she may have been entirely innocent of any fault whatsoever. We know that Moses had marriage problems inasmuch as his Midianite wife Zipporah was deeply resentful over the rite of circumcision (Ex.4:25). Indeed, the problems seem to have been so bad between the two that in the end Moses was obliged to send her and their two sons back to Midian with his father-in-law (Ex.18:2). One is reminded here of the trouble between Abraham and Hagar whom he was forced to send away together with their son Ishmael ... quite the opposite image the monogamy-only-promoting Hollywood movie tried to convey in which Zipporah was portrayed as the ideal monogamous wife. However, we must note in passing that Gershom, Moses' eldest son, did not have an inheritance status remotely comparable to that of Ishmael, for we find him listed amongst the Levites in 1 Chronicles 23:14-15, as would be expected since this was his father's tribe. His sons do not figure latterly in Israelite history.
The kind of culture Moses' first wife came from
Yahweh's intervention against Miriam by striking her with leprosy can be seen in many ways. Though the Ethiopian woman was the excuse, the main cause of the punishment against her was her rebellion against Moses' authority. It was primarily an issue of patriarchy vs. matriarchy, or at least some kind of demand for coheadship. Miriam had been greatly blessed of Yahweh in many ways and had misinterpreted this as an indication that she enjoyed a position of authority in Israel. She tried to bring her brother Aaron into the conspiracy by claiming that they had 'equal rights' to their senior brother. But she was wrong, and indeed demonstrated a latent boastful and imposing feministic spirit contrary to that expected by Yahweh.
Though this episode did not, I think, have anything directly to do with polygamy there can be no doubt that Yahweh's vindication of Moses' first marriage was in a secondary sense a vindication of polygamy as we are nowhere told that his marriage to the Midianite woman was in any way wrong. Or one could argue that Yahweh wasn't actually interested in the marriage issues at all in this instance but in extirpating Egyptian matriarchy and a latent spirit of Jezebel. One thing we can be sure about is that there is not the slightest hint in the family history of Moses of Yahweh's disapproval of polygamy or His trying to set up a monogamy model. Such concepts are entirely absent from the accounts and from tradition, and contrary to the generally pro-polygamy attitudes of the chosen nation.
So strong was the polygamous tradition amongst the Hebrews that it was not until 1020 A.D. that the Western Talmudic Jews finally banned it. This was done at the Rabbinical Council of Worms (an appropriate name!) when Rabbi Gershom ben Yehuda of Mainz (Mayence, Germany) persuaded the rabbis to issue an edict putting a 'legal' stop to polygamy. That they had no authority scripturally to do this goes without saying since by New Testament definition they were not, in any case, Israel any longer -- those Hebrews who reject the Messiah having their branches cut off, which they had done. But even this aside -- even if they had still been bona fide covenant Israelites in Yahweh's eyes - their state of apostacy from even the then defunct ceremonial arm of the Law of Moses was clear because of their attitude to the Bible: "My son, be more careful in the observance of the words of the Scribes than in the words of the Torah [Old Testament]" (Talmud Erubin 21b, Soncino edition). They, liken the monogamy-only Christians/Messianics of our day, loved their scribal traditions more than the truth.
 The Second Exodus Lineup: Moses and the Twelve Moses-Men