Easy example why complex numbers are cool I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary.
The best example would be possible to explain rigourously and also be clearly important in a daily day sense.
I.e. complex Laplace transform has applications in pricing of options in mathematical finance which is somewhat easy to sell as important, but impossible to explain the details of.
It is easy to say: Then we can generalise the square root! - but it is harder to argue why that makes any difference in the real world.
The question has edited the wording cool out of it replaced with a description to stop it from being opinion based. I hope it helps :)

At the very beginning of the study of geometry, one learns to place points in a Cartesian frame, often an orthonormal frame.
The set of real numbers (or real line) is denoted $\mathbb{R}$; there is no distinction to be made between points $(x,y)$ of the euclidean plane $\mathbb{R}\times\mathbb{R}$ and complex numbers $x+iy$; if we note $\mathbb{C}$ the set of complex numbers instead of $\mathbb{R}\times\mathbb{R} $ or $\mathbb{R}^2$, it is just to remember that in addition to the addition of the vectors in $\mathbb{R}\times\mathbb{R} $, there is defined a multiplication which makes $\mathbb{C}$ a field.
Complex numbers are necessary because, without them,  no geometry, no algebra, no mathematics.
