Good idea. Although, from the little exploration I did with examples for small n, given the low stopping times, the behaviour was very boring. Nothing necessarily interesting. But then again, I didn't explore it exhaustively.
Long and/or large excursions can happen even for small n! As mentioned at https://en.wikipedia.org/wiki/Collatz_conjecture#Empirical_d... , for example, 27 meanders for quite a while before reaching the inevitable cycle.
Always interesting to try to visualize something though.
Ps -- I implemented hashlife one time. Still amazed someone came up with that algorithm