In algebra II, USA, we teach our students complex numbers. However, after algebra II, they never use complex numbers until pretty much complex analysis. The whole point of teaching them complex numbers is to find the roots of polynomials... but, that's all we ever do with them. Sure, we do some algebraic manipulation of them, just to get a feel, but that's it. Nothing ever too deep, and often times there isn't even a geometric understanding of them.
And by the time students ever reach a point where they need to use complex numbers... it'll be very far in the future.
As far as real world applications, it'll probably be finding the real roots of something for anyone who doesn't take complex analysis or similar courses that deal with complex numbers. And at that point, you probably don't sit down and factor your polynomial. I would think most would either see it as a quadratic or go straight to a calculator (one that can solve such things with complex numbers or numerical methods)
So, my question is if its really worth it teaching students about complex numbers. Most students will forget about complex numbers even, as complex analysis is usually a far way off, if they ever get there.
Maybe to be more specific,
Why do we teach students about complex numbers if most will never reach a course that uses them? When do laymen use complex numbers in real world applications?