[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 thing properties that the normal ℂ has this guy will also have.

Now, this pair notation lacks the wow factor that i has, like "Whaaaaaat!!! Square root of -1!!! Get your eyes away from my daughter you crazy mathematician!!!". Or "Huh? What do you mean an Imaginary number????"

But, maybe more crucial, the equations would look less mysterious and attractive without i all over the place; for example this doesn't look enchanting at all

##e^{[0,1]*[\pi,0]} = [-1,0]##

So, is it possible that mathematicians like i just because it leads to compact, attractive and mysterious equations -- that is, the choice of using it all the time is just basically aesthetic?