Is there a construction of (C, +, x) that sets i^2 = -1 as a definition rather than a property?

I often have arguments with people because I disagree that there is a simple construction of (C, +, x) that can fully define the product and reliy solely on i^2 = -1.

There are multiple ways to construct it but the one that is to me the most basic is:

Let set a non real number i, and construct C = {a+bi, (a,b) in R^2}

w the following conventions:

• 0+bi/a+0i is just written bi/a and 1*i is written i

Let's define (C, +, x) such that;

• a + bi + c + di = a+c + (b+d)i

• (a + bi) x (c + di) = ac-bd + (ad+bc)i

such operations define a field (straightforward to verify)

Then as a property, taking a = c = 0 and b = d = 1 we have i^2 = -1, putting it in the definition would hence just be redundant.

Usually I see people saying that you just set i^2 = -1 and that the definition of the canonical multiplication is induced. I disagree, it would be the case if such a construction would guarantee uniqueness of the field product.

Here a counterexample:

let f such that f(2i) = 3i, f(3i) = 2i and f(z) = z otherwise.

let's define x as z1 x z2 = f^-1 (f(z1) f(z2))

this product conserves field properties, verifies i^2 = -1 however (2i)^2 = -9 != -4.

My point is that as soon as you define the canonical product which therefore needs to be defined because of the non uniqueness i^2 = -1 becomes a property.

My question is the title