The simple fact you wanted to write this:

i=0+1*i

Makes i a number. Since * is a binary operator in your space, i needs to be a number for 1*i to make any sense.

Similarly, if = is to be a binary relation in your space, i needs to be a number for i={anything} to make sense.

Comparing i with a unary operator like - shows the difference:

i*i=-1 makes perfect sense

-*-=???? does not make sense

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