[roywiggins]

Imaginary numbers were useful as soon as they were invented. They are algebraically closed, unlike the real numbers, and are required for the fundamental theorem of algebra.

https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_algeb...

The first use of negative roots was to find the (real) roots of certain polynomials. They were considered a mathemathical hack: useful, but specious on their own. And then it turned out that if you take them on their own terms, they're fantastically useful.