Numbers 12:1 states that Moses had an ethiopian wife, and this caused controversy. This occured between leaving Egypt and the first attempt to enter Canaan, a short period during which Zipporah is also mentioned as being still married to him (Exodus 18). So just from scripture alone we can deduce that Moses had two wives simultaneously.
But much more information about Moses is given by Josephus, and when we read his account also this all becomes much clearer.
Prior to his banishment, Moses became a very prominent figure in Egypt. When Ethiopia invaded Egypt, and Egypt was rapidly being defeated, Moses was appointed the commander of the army. He turned the entire war around through some shrewd maneouvers, and defeated the Ethiopians. The final seige against their capital was ended when the daughter of the Ethiopian king brokered a surrender, partly involving giving herself to Moses as a wife - her idea, she saw him from the wall and was very attracted to him. (
Antiquities of the Jews, book 2, chapter 10).
Which means that she was Moses' first recorded wife. After his banishment he took Zipporah to wife as outlined in scripture. Then when his Ethiopian wife is mentioned in Numbers as causing a controversy, and therefore still around (if she'd died many years before there could have been no controversy), we find that the entire time he has been married to Zipporah he has been a polygamist! We don't know if this Ethiopian wife accompanied him to Midian or stayed in Egypt though in a "long-term distant relationship".
We also read of another father-in-law to Moses, Hobab the Kenite (Judges 4:11), a different name and family to Jethro, Zipporah's father. This strongly indicates a third wife, though we know nothing about her so cannot be certain that she was married to him at the same time as the others.
So it is very clear that Moses was a polygamist with at least two wives, and possibly more.