I heard an interesting, but confusing, chumrah observed by Rabbi Pesach Levi of Lakewood. He said one should get dressed in one's bathroom because one should not appear naked in front of one's personal angel, and because angels do not accompany a Jew into the bathroom. That made me wonder whether we should be concerned about being modest before our angels when we go to the mikvah. Does anyone know the source for this opinion and can they explain how it might apply to the mikvah?
While I cannot corroborate the idea that the angels do not accompany us to the bathroom, I know that there is support both for a similar stringency, and for its non-applicability in the Mikvah.
One should be modest and never appear naked, even when getting dressed; to that end, one should get dressed under one's covers.
When entering a Mikvah or public bath, or going into a stream to bathe, men should specifically not cover themselves, lest it appear that they are embarrassed of, and trying to hide, their מקום המילה.
However, #2 above is only upon entering the waters, when one's back is towards anyone else present. Upon exiting the waters, though, when one is actually facing towards others, he should try to cover himself. In other words, the primary goal is modesty, but at the same time one should not appear to be deliberately hiding one's ברית.
(H/t @Avrohom Yitzchok)