Does there exist the idea of "RP-complete", like NP-complete?
Yes: the concept of "X-complete under Y-reductions" exists for any complexity class&nbsp;X and any class&nbsp;Y of reductions.  However, there may or may not be any complete problems under this definition, depending on what X and&nbsp;Y are. For example, it is well-known that NP has complete problems under polynomial-time, logspace and even first-order reductions, but it does not have complete problems under linear-time reductions as this would violate the time hierarchy theorem.

As far as I can see, it is open whether there is a class of reductions under which RP has complete problems. The issue is that it is a so-called semantic class: it is defined by a non-computable set of Turing machines, namely polynomial-time randomized Turing machines with the undecidable requirement that, for every input, either every path rejects or at least half accept. See this question for more details on the issues surrounding complete problems for semantic classes.