i is a complex number, complex numbers are of the form real + i*real... Don't you see the recursive definition ? Same with 0 and 1 they are not numbers until you can actually define numbers, using 0 and 1

  i*i=-1 makes perfect sense

There's no issue with recursive definitions. That's how arithmetic was original formalized by Peano's axioms [1].

[1] https://en.wikipedia.org/wiki/Peano_axioms