I've been in touch with my current PI's phd advisor, and we seem to get along really well - he's supposedly a great advisor, at a school I really want, and in a topic I'm really interested in, and I've been very extensively trained in a lot of things that he does because I'm working in something very similar now (and, obviously, under a former student of his). Also, of course, we have the connection of my current PI. They still publish together a lot (~20 years later), so it's one of those close relationships that continues long after graduation, which is nice to see.
My only concern is that it feels a bit like intellectual inbreeding - I know that my current PI was vastly influenced by this guy, and I've already been influenced by my current PI - am I handicapping myself by not giving myself a diversity of viewpoints in my mentors? Will people who know that they're very tied to each other (it's a bit of a running joke in the subfield that this guy's old students publish together or with him a lot) raise eyebrows in future job searches? Say, if I'm looking for a postdoc?
On the other hand, it is what I want to do, and I could probably productive on the earlier side..
It's hard to get good advice on this here - my PI is obviously biased, and his students/postdocs all have strong opinions, and the other professors here respect my PI too much to disagree.
[Note: I'm an undergrad. I've been working with my PI since the beginning of freshman year, and we have a total of three papers (2 first and one second author) plus one review (1st author) either in the works, published, or in the middle of the submission process]