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
This is one definition of i. Or you could geometrically say i is the orthogonal unit vector in the (real,real) plane where you define multiplication as multiplying length and adding angles
i is also a quaternion. So by this logic we could say complex numbers are made up of quaternions. But we don’t say such things because they wouldn’t be a good mental model of what we want to talk about.
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