Absolutely. Otherwise, all compilers would be written in Assmebly. Most compilers used today are written in C and the compiler for C (GCC) is written in C itself.
The C language was also written by Ken Thomson, because he was unhappy with the BCPL language, so the compiler for C was originally written in B. So on and So forth.
More Recently, the initial compiler for Rust was written in OCaml but is now written in Rust itself.
https://stackoverflow.com/questions/27513234/in-which-langua...
Its been a while, and I think it was gcc 7.x timeframe, and IIRC, you had to have at a min, gcc 4.x available to bootstrap, and it took something like 3 separate bootstrap iterations to get the final compilers was like 4-6 hours if memory serves.
There is certainly a major milestone in each new language though, when it becomes feature complete enough to write its own compiler.
A better compiler can compile itself more efficiently, same for CPU/chip layout systems.