Is a naturally self-impregnated androgynus's child a mamzer?
A mamzer is the result of a relationship prohibited to the point of spiritual excision (Mishna, Yevamos 4:13). There is no prohibition of any severity for relations with one's self. Every act of normal relations that is prohibited has a verse and explanation of what relationship between those two people forbids their relations. Homosexual relations, which can be (hypothetically) violated with one's self, is prohibited without any consideration of who the individuals involved are, and therefore one's self could be (hypothetically) included. However, relations with the female part of an androgynous is not considered homosexual relations (Rambam Isurei Biah 1:15). There is therefore no scriptual basis for such a prohibition - it does not fit into any of the 15 relations for which the punishment is spiritual excision.
The gemara in Sanhedrin 55a is indicative of this -
בעא מיניה רב אחדבוי בר אמי מרב ששת המערה בעצמו מהו אמר ליה קבסתן אמר רב אשי מאי תיבעי לך בקושי לא משכחת לה כי משכחת לה במשמש מת למאן דאמר משמש מת בעריות פטור הכא פטור ולמאן דאמר חייב הכא מיחייב תרתי מיחייב אשוכב ומיחייב אנשכב
Summary: Rav Achdavuy: What is the halacha of one who commits sodomy with themselves? Rav Sheshes: Stop bothering me with ridiculous questions! (See Rashi.) Rav Ashi: This case cannot exist with an erection. You can only ask without an erection. According to the opinion that there is a prohibition without an erection, it violates two prohibitions. According to the opinion that there is no prohibition without an erection, there is no prohibition. (Halacha follows the latter opinion.)
So the Gemara was unable to find a case of someone violating themselves with an erection. If the case of this post's question provided such a case, Rav Sheshes should not have scolded Rav Achdavuy and Rav Ashi should not have needed to resort to an opinion that is not normative halacha. One is forced to conclude that either such a case is impossible, and an androgynous could not impregnate themselves naturally, or such a case is not an erva, as there is no verse in the Torah prohibiting the "relative" of one's self. As opposed to the case the Gemara did ask about, in which the prohibition is sodomy which has nothing to do with the relationship between the individuals.
Therefore, the case of the question either does not exist or is not prohibited by the Torah (as a forbidden act of relations).