The nicest elementary illustration I know of the relevance of complex numbers to calculus is its link to radius of convergence, which student learn how to compute by various tests, but more mechanically than conceptually. The series for $1/(1-x)$, $\log(1+x)$, and $\sqrt{1+x}$ have radius of convergence 1 and we can see why: there's a problem at one of the endpoints of the interval of convergence (the function blows up or it's not differentiable). However, the function $1/(1+x^2)$ is nice and smooth on the whole real line with no apparent problems, but its radius of convergence at the origin is 1. From the viewpoint of real analysis this is strange: why does the series stop converging? Well, if you look at distance 1 in the complex plane...
More generally, you can tell them that for any rational function $p(x)/q(x)$, in reduced form, the radius of convergence of this function at a number $a$ (on the real line) is precisely the distance from $a$ to the nearest zero of the denominator, even if that nearest zero is not real. In other words, to really understand the radius of convergence in a general sense you have to work over the complex numbers. (Yes, there are subtle distinctions between smoothness and analyticity which are relevant here, but you don't have to discuss that to get across the idea.)
Similarly, the function $x/(e^x-1)$ is smooth but has a finite radius of convergence $2\pi$ (not sure if you can make this numerically apparent). Again, on the real line the reason for this is not visible, but in the complex plane there is a good explanation.
