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 :)

By far the coolest thing about complex numbers is that nature embraces them. 
Throw a die. Since Newton we know how non-linear dynamics laws are supposed to govern the probabilistic outcomes. Now we know better. We understand nature at a more fundamental level. And at that level we work with complex amplitudes, the 2-norm of which corresponds to observed probabilities. The truly amazing thing is that the dynamics laws that govern the complex amplitude dynamics are entirely linear. 
That's right, all the complexity and chaotic behavior observed in real life, exemplified by the throw of a die, results from fundamentally simple linear evolution laws. And for these laws to work and to produce the richness of behaviors we observe, you have to introduce a complex number description of nature. 
Complex numbers are a necessity. Without them, you can't understand the physical world.
