Why not?

But seriously. A better question is actually: "Why do we need to work exclusively in a real number?"

I invented a new kind of mathematical object. I call it a "Weird Pair", or WP for short. It's represented as a pair of real numbers [a, b]. Two WPs are equal if each number is equal. Let's define a few operations we can do to a weird pair:
[a, b] + [c, d] = [a + c, b + d]
[a, b] * [c, d] = [a * c - b * d, a * d + b * c]

Hmm. Interesting. What is this useful for? Here is one of the possible uses. Let's interpret the Weird Pair as a regular pair. Looks like we can represent a point in a R^(2) space. The addition looks like a translation formula. But what does the multiplication do? Notice that the formula of rotating a point around the point of origin is (x cos θ - y sin θ, x sin θ + y cos θ), so, if the second WP is [r cos θ, r sin θ], we have invented a way to describe scaling and rotation.

Now, I wonder what is the sample operation of WP.
[0, 1] * [0, 1] = [0*0 - 1*1, 0*1 + 1*0] = [-1, 0].
Hmm, interesting. Also, what if I only care about the first element:
[a, 0] * [b, 0] = [a*b - 0*0, a*0 + b*0] = [ab, 0]. [a, 0] + [b, 0] = [a + b, 0].
So, I can treat a WP with the second element zero as a regular number. Wait a minute.

Whoops, I accidentally reinvented an imaginary number.