And FWIW the author of FFTW is a big Julia contributor / package author nowadays. A long time ago (2014) he did write a Julia-based version of FFTW:
https://github.com/JuliaLang/julia/pull/6193#issuecomment-61...
This was back before Julia could do auto-SIMD optimizations, and it got quite close to FFTW without SIMD. He mentioned somewhere that modern Julia should be able to get very close to FFTW with SIMD, given that Julia now has better inlining heuristics, interprocedural optimizations, automatic SIMD, etc. (2014 was the stone age for Julia). If you read the rest of the thread, most of the discussion was about a standard library system so FFTW could be moved out. With Julia v1.0 FFTW is now in the standard library, which gives room for a pure Julia FFT to be standard. So this should get revived soon and we will have a beautiful generic FFTW algorithm. I can't wait :)