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 :)

Complex numbers can be used to give a very simple and short proof that the sum of the angles of a triangle is 180 degrees. Construct a triangle in the plane whose vertices are complex numbers z1.z2 and z3 in the plane. 

Recall that the angle $\theta$ at a vertex like $ABC$ is the angular part $\theta$ of $(C-B)/(A-B) = r\exp(i\theta)$.  It is written $\theta = \arg(r\exp(i\theta))$ and is defined only modulo $2\pi$, although conventionally a representative in the interval $(-\pi, \pi]$ is taken.
Now recall that $\arg(z w) =\arg(z) + \arg(w)$ for any nonzero complex numbers $z$ and $w$: this follows immediately from the multiplicative property of the exponential, $\exp(i\theta)\exp(i\phi) = \exp(i(\theta + \phi))$.
The sum of the three angles at vertices $Z1$, $Z2$, and $Z3$ is obtained merely by rearranging the product of three numbers in the denominator of a fraction. So now: 
$$\eqalign{\arg\frac{Z2-Z1}{Z3-Z1} + \arg\frac{Z3-Z2}{Z1-Z2}+\arg\frac{Z1-Z3}{Z2-Z3} &= \arg\left(\frac{Z2-Z1}{Z3-Z1}\frac{Z1-Z3}{Z2-Z3}\right) \\
&= \arg\left(\frac{Z2-Z1}{Z1-Z2}\frac{Z3-Z2}{Z2-Z3}\frac{Z1-Z3}{Z3-Z1}\right) \\
&= \arg\left(-1^3\right) = \arg(-1) \\
&= \pi.
}$$
Ligget se. I'll always remember Ravi Kulkarni showing us this trick at the end of one of his lectures and quoting his granddaughter, "Cool." I hope your students agree-especially if they've just been tortured with the painfully tedious classical proof of the result. 
