Did Moses marry a second time? Maybe his first wife died or maybe we don't know for sure? I am wondering if the wife that Miriam and Aaron complained about in parshas Beha'alosecha (Bamidbar 12:1) is actually the same first wife just being described differently?
The Yalkut Shimoni in parshas Shemos here discusses Moshe's time in Ethiopia, and records there that after Moshe successfully helped the Cushites conquer a very fortified city:
they placed him on the throne and placed the royal crown on his head, and also the Cushite noble woman (the wife of the previous dead king, as it mentions earlier) they gave to him for a wife. But Moshe feared the God of his forefathers and did not sleep with her, because he remembered the oath which Avraham made his servant Eliezer swear, saying: Do not take a wife for my son from the daughters of Canaan. And so too did Yitzchok when Yaakov fled because of Eisav - he commanded him and said to him: Do not marry from the descendants of Ham, because we remember that God gave the descendants of Ham as servants to the descendants of Shem and to the descendants of Yefes.
The Midrash continues that he ruled there for forty years, after which time his 'wife' complained to the ministers and the people:
"Behold, for forty years this one has ruled over Cush, but he has never slept with me, and he has never worshiped our idols!"
She advised them to instead make her son the king, and they agreed to this and sent Moshe away with a lot of gifts and with great honor.
Thus, according to this Midrash, Moshe had been 'married' before he came to Midian and married Tzipporah, (although since he never consummated the first 'marriage' it could be argued that it should not be considered a real marriage) and the Targum Yonasan and the Rashbam on the posuk in parshas Beha'alosecha use this Midrash to explain that the Cushite woman mentioned to in the posuk there refers to this first Cushite wife.
Rav Shimshon Rafael Hirsch (among others) points out that the term used is "Al Odos" which would normally be translated as "on behalf of". Rav Hirsch also states that "Isha Cushis" is not being literal but is a complaint on behalf of Tzipporah that Moshe is treating her as if she was as separated or far apart from him as the Cushiyim are in the idiom in Amos 9:7. A marriage in which the partners live sexually apart is objected to and improper.
Rav Hirsch (and others) state that this does not refer to the medrash of Moshe "marrying" the queen of Cush before he went to Midian. I assert that Rav Hirsch would have mentioned it if there were other wives mentioned in other places. Thus, the answer to your question would be that this is indeed Moshe having only marries Tziporah who was his only wife.
There is a midrash that Moses was briefly the king of Ethiopia when he helped a deposed king regain his throne and the king died. He was given the dead king's wife but did not sleep with her and put his sword between their beds.
Miriam is referencing how Moses no longer sleeps with Tzipporah (due to his always being on-call to for God)
apparanly Josephus also recounts this story, but I have heard it in a Jewish context. http://fontes.lstc.edu/~rklein/Documents/did_moses_marry_a_cushite.htm
Besides all the other references, you also have the Targum Yonasan on that Passuk:
וְאִשְׁתְּעִיוּ מִרְיָם וְאַהֲרן בְּמשֶׁה פִּתְגָמִין דְלָא מְהַגְנִין עַל עֵיסַק אִתְּתָא כּוּשְׁיָיתָא דְאַסְבוֹהִי כוּשָׁאֵי לְמשֶׁה בְּמֵיעַרְקֵיהּ מִן קֳדָם פַּרְעה וְרִיחְקָהּ אֲרוּם לְאִיתָא אַסְבוֹהִי יַת מַלְכְּתָא דְכוּשׁ וְרָחִיק מִינָהּ
And Miriam and Aaron spoke inappropriately about the Kushite woman, for a Kushite was married to Moshe when he escaped from Pharaoh, and he distanced (divorced?) her, for as a wife the Queen of Kush married him and he distanced her.
Not clear to me if the Queen forced /suggested to him to get married, or if she was the bride in question.