Would it be fair to state that most if not all of the practical applications of complex numbers stem from the relationship between $\pi$, $e$, and $i$ described in Euler's identity ($e^{i \pi} +1 = 0$)?
I've found that some of the most common practical applications of complex numbers include the study of brain waves, electrical engineering, and wireless technologies such as cell phones and radar. It appears as though all of the practical applications of complex numbers are in some way related to waves and oscillation, and therefore to $\pi$.
Clarification: I'm looking for a way I can boil down the practical/real-world applications (thus leaving out their utility in providing completeness to the fundamental theorem of algebra) of complex numbers to someone who just now for the first time was introduced to the idea that $\sqrt{-1}$ exists, and who has only a very basic understanding of trig, and no knowledge of polar coordinates or quantum theory. If the statement "most of the practical applications of complex numbers stem from the relationship between $\pi$, $e$, and $i$ described in Euler's identity ($e^{i \pi} +1 = 0$)" is a true statement, and no simpler way exists to concisely describe to practical usefulness of complex numbers, that's what I'm going to go with.
This is like saying all practical uses of mathematics come from numbers. Obviously they are at the core, but there is so much more you can do with math. $\pi$ shows up in complex numbers because of their polar form. Every complex number $z$ can be written as \begin{equation*} z = r(\cos{\theta} + i\sin{\theta}) \end{equation*} which in turn can be written as \begin{equation*} z = re^{i\theta}. \end{equation*} The formula you wrote is a special case of this, when $\theta=\pi$ and $r=1$.