Looking at the graph of the prime indexed primes on oeis and the graph of the primes, it seems that the first graph has small jumps exactly at the same positions as the second.
It's no surprise that both graphs show small jumps, but I'm surprised that they seem to appear at the same indices. (e.g. the small jumps shortly before 200 or around 150). They seem very similiar in general, just with different scale on the y-axis. What could explain this strong similarity?
After playing around with this a bit I think that it's not surprising at all to see the same pattern again.
If we repeat the process of prime indexing the primes, the initial irregularities get amplified because these sequences grow so fast.
In the plots beneath, in each iteration the values of the previous sequence are used as the indices for the primes. The first iteration uses the primes as indices (A006450).
While a small jump in the plot of the primes could mean a difference of say 6, a small jump in a plot of the 6th iteration would correspond to a difference of thousands/millions, so that any other irregularities/patters of prime numbers of that magnitude are tiny compared the preceeding/following values in that sequence.
A fun thing I noticed while plotting this: the more iterations, the more of the first values of the resulting sequence will be in A007097 (Primeth recurrence: a(n+1) = a(n)-th prime).