I know of some people who are particular not to go to the mikveh together with a close relative (father-son , brothers, and even brother-in-laws).
On the other hand I also have seen Chassidim and Yirei Shomayim who are not particular about this (including a father going in with his young son for Chinuch).
What are the issues involved here? AFAIK this a matter of halacha as I have seen the Remah in Shulchan Aruch (Aven HaEzer Ch 23 Halacha 6) referces this issue.