If the students have had a first course in differential equations, tell them to solve the system

$$x'(t) = -y(t)$$ $$y'(t) = x(t).$$

This is the equation of motion for a particle whose velocity vector is always perpendicular to its displacement. Explain why this is the same thing as

$$(x(t) + iy(t))' = i(x(t) + iy(t))$$

hence that, with the right initial conditions, the solution is

$$x(t) + iy(t) = e^{it}.$$

On the other hand, a particle whose velocity vector is always perpendicular to its displacement travels in a circle. Hence, again with the right initial conditions, $x(t) = \cos t, y(t) = \sin t$. (At this point you might reiterate that complex numbers are real $2 \times 2$ matrices, assuming they have seen this method for solving systems of differential equations.)