They also have a defined multiplication operation consistent with how it works on ordinary numbers. And it's not just multiplying each number separately.

A lot of math works better on them. For example, all n-degree polynomials have exactly n roots, and all smooth complex functions have a polynomial approximation at every point. Neither is true on the reals.

Quantum mechanics could possibly work with pairs of real numbers, but it would be unclear what each one means on their own. Treating a probability amplitude as a single number is more satisfying in a lot of ways.

That they don't exist is still a position you could take, but so is the opposite.

I totally get your point, and sometimes it seems like that. Why not just use a coordinate system? Because in some applications the complex roots of equations is relevant.

If you square an imaginary number, it's no longer an imaginary number. Now it's a real number! That's not something you can accomplish with something like a pair of numbers alone.

Because the second number has special rules and a unit. It's not just a pair of numbers, though it can be represented through a pair of numbers (really helpful for computing).