Say that I define a set of pairs called ℂ, such that
[a,b] ∈ ℂ iff
a ∈ ℝ, b ∈ ℝ,
[a,b]+[c,d] = [a+c,b+d]
[a,b]*[c,d] = [ac-bd, ad+bc]
Then this has exactly the same properties of a+bi, does it not? You can write any equation that uses i exactly the same way with those pairs, so all interesting...