Jon Pretty

are you thinking of something like https://www2.clarku.edu/faculty/djoyce/complex/cubic.html … where in solving cubics you need complex numbers but you have real solutions? counterpoint is lots of equations have complex solutions and complex numbers do more for us besides

Yes, this is the idea. Complex numbers were introduced as a means of solving cubic equations which could not be evaluated without them, but the solutions eliminate all the complex terms from the answer. The idea is that if they were not eliminated, the answer wouldn't be useful.

I think this fact is interesting, I think it shows the utility of complex numbers (you can solve things you could not do before, interesting special case is when you go via complex to get real solutions) But this is not all they do

It might be that over time, the new concept proves itself so useful for this purpose that it's established as something so fundamental that the original system (the natural numbers without negative numbers, or the reals without complex numbers) seems less useful.

Do you still assert they only exist to be "eliminated"?

I agree with @berenguel, more like they subsume the existing thing than are later eliminated (though I am really struggling to understand if "eliminated" means removed from mathematics or means it has this property you seem to like to see expressions have sometimes )

I certainly didn't mean that complex numbers should be removed from mathematics. But if we have a problem that exists exclusively in the reals, we can't make use of a solution in the complex numbers. But we might use complex numbers as a tool to reach a real solution.

Oh but that’s it: we can! Real-analytic problems might have no solution if restricted to the reals! You need tools from complex analysis