If it’s complexity you want, listen to J.S. Bach’s Concerto For 4 Harpsichords and Orchestra. Each harpsichord plays a different melody; just trying to keep those 4 threads separate will keep your mind busy! And those 4 parts are played over an insane string of chords, the string played at breakneck speed. A lot of other Baroque compositions have a mind-boggling progression of chords through which the musicians must traverse. And modulations (key changes), not to mention the counterpoint parts in the Fugue form. The hardest thing I ever had to learn to do was sing one part in a Fugue-based Pop song, when I was recording with a songwriter who was a music major at The University of California at Riverside, known for it’s excellent music department. It was he who turned me on to J.S. Bach. By the way, his favorite Pop music songwriter was Brian Wilson.
"Classical music is it’s own genre." Well yeah, of course. I agree, you can’t compare it with Blues-based music. But it was you who said "Jazz is the most complex music ever created." You didn’t say the most complex Blues-based/non-Classical music.
As for improvising, you apparently aren’t aware that a lot of Baroque compositions are written with sections requiring the harpsichordist to provide "ornamentation", the baroque term for improvisation. Keith Jarrett is one pianist equally capable in both Classical and Jazz musics; I have a bunch of his Classical recordings.
Terry Adams is one Rock ’n’ Roll keyboardist (piano, clavinet) who uses Jazz influences (Thelonious Monk, Sun Ra) is his band’s (NRBQ---an acronym for New Rhythm & Blues Quartet. Still going strong after 55 years!) songs, and does so with a great sense of humour. The best band in Rock ’n’ Roll!