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
Most of the existing answers here cite interpretations about a woman that Moses had previously married in Ethiopia. However, they do not explain why past events in Ethiopia suddenly became a hot topic for Miriam to talk about.
The "classic" interpretation in some Midrashim and popularized in commentaries like Rashi is that they were talking about Moses's wife Tziporah. Moses had separated from her on account of his prophetic duties, and Miriam was criticizing him for this. In this interpretation Moses's wife is referred to as a Cushite as a euphemism for some quality, such as her beauty or good deeds. The reason why Miriam brought it up now is that she had just found out about it in response to the prophecies of Eldad and Meidad.
One issue with this is that there is not the slightest hint of this in the Scriptural text; indeed separating from a wife is the very opposite of the term לקח used in the verse. R. Joseph Ibn Kaspi discusses this point at length in his commentary, going so far as to accuse all the previous commentators of giving free license to interpret the Torah to mean the exact opposite of what it means. He says, for example, that using this methodology one might say that when the verse says "you shall love the Lord your God" it in fact means that you should hate God, or that when Scripture says "and you the Lord took" it in fact means that God abandoned them. Ibn Kaspi's ire is twofold – this interpretation requires inserting a non-existent clause that negates what Scripture says (i.e. that they separated which negates that he took her) and it also interprets the word Cushite as meaning the opposite of what it means.
After leveling these accusations against his predecessors, Ibn Kaspi provides his own simple interpretation:
לכן אומר אני כי פירוש מה שכתוב בכאן לפי המובן ביאור הכרחי מלשון העברי כי כושית היא אשה מארץ כוש ופירוש כי אשה כושית לקח החזיק כי כן פירוש שורש הלמ"ד והקו"ף והחי"ת והפך זה שרש העי"ן והזי"ן והבי"ת וכל זה הסכמת הלשון והיה הענין כן כי משה אחרי נשאו צפורה לקח אשה כושית על צפורה לסבה שידע הוא ע"ה ואין לשאול טעמים בפעולותיו כי בודאי מחכמה עשה זה והנה לא ידענו הזמן המוגבל שנשא זאת אם עתה בנסעם אם לפנים זמן מה כי לא נזכר בתורה עדיין כי גם לא נכתבו בתורה כמה ענינים עברו אז אבל בעבור שלא נזכר זה עתה בעבור שלא נבהל באומרו על אודות האשה הכושית אשר לקח בעבור שלא שמענו זה מעולם אמר אחר זה כי אשה כושית לקח כאלו אמר דעו שאשה כושית לקח ואם לא נזכר עדיין ועליה דברו אלה כי בעבור העלם מהם סבת זאת ההוספה בלקיחת האשה הזאת כי אולי חלתה צפורה בבית הסתרים ואולי מרדה בו ואולי סבה אחרת לא ידענו מה ואפשר שידעו אלה סבתה אבל נעלם מהם סוד ההכרח והחכמה בזה ולכן דברו על משה בהוסיפו אשה על אשה כי די לאחת לאיש השלם
Therefore I say that the explanation of that which is written here as is understood is an explanation forced by the Hebrew language. For a Cushite is a woman from the land of Cush, and "for a Cushite woman he had taken" means that he possessed [her], as that is the meaning of the root ל-ק-ח and the opposite of this is the root ע-ז-ב, and all this is the agreement of the language. And the matter was such because Moses – after marrying Tziporah – took a Cushite woman in addition to Tziporah, for some reason known to him (peace be unto him). And we should not seek reasons for his actions, because he certainly did it from wisdom.
And behold we do not know the precise time that he married this [Cushite woman] – if it was now while they were traveling, or slightly earlier – since the Torah had not previously mentioned it for the Torah [indeed] does not mention many things that occurred then. But since it had not [yet] mentioned this now, [therefore] in order that we should not be startled when it says "regarding the Cushite woman whom he had taken" – since we have never heard of this – it [thus] says after this "for a Cushite woman he had taken", as if to say "you should know that he had taken a Cushite woman even if it hadn't been mentioned yet".
And it was about her that these spoke, because due to their ignorance of the reason for this addition of taking this woman – for perhaps Tziporah had fallen ill in private, or perhaps she had rebelled against him, or perhaps some other reason which we do not know. And it is possible that they did know the reason but were ignorant of the secret of the force and wisdom of the reason. And they therefore spoke about Moses when he took a wife in addition to his wife, for it is sufficient for a wholesome man to have one [wife].
R. Joseph Bekhor Shor has a somewhat similar approach:
לפי הפשט שהיו אומרים וכי לא מצא משה אשה מבנות ישראל שיקח לו לאשה שהלך לקחת לו מבנות הכושים שהם ערלים וכי בשביל שהקב"ה מדבר עמו מתגאה שאינו רוצה לישא אשה מבנות ישראל שבקש אשה במרחק
According to the plain meaning: for they were saying "could Moses not find a woman from the daughters of Israel to take as a wife, that he had to go take from the daughters of the Cushites who are uncircumcized? Because God speaks with him he became haughty that he does not want to marry a woman from the daughters of Israel, that he sought a woman from afar?"
So Miriam and Aaron were simply upset that Moses apparently thought himself too great to marry a Jewish woman. He then clarifies that this was a new wife, not Tziporah:
לפי שלא מצינו במקום אחר שמשה לקח אשה כושית הגיד לך הכתוב כי בודאי אשה כושית לקח ועל צפורה לא דברו שלא היתה מישראל כי אז אנוס היה מפני שהיה בורח ולא יכול לבא במצרים בתוך בני ישראל
Because we do not find anywhere that Moses took a Cushite woman, Scripture tells us that he certainly took a Cushite woman. But about Tziporah they did not speak [even though] she was not from Israel, because there he was forced because he was on the run and could not come to Egypt amongst Israel.
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.