*(Expanding my comment into an answer. Complex roots, integrals, and trig identities are* preaching to the choir. *If you consider those things "cool", then you probably think the same about the concept of complex numbers themselves, and therefore do not need any convincing. Contrapositively, people who do not* yet *see the intrinsic beauty of the complex plane are unlikely to be persuaded by other examples of abstract math.)*

Here's a weird name that most students have never heard of: Quadrature Phase Shift Keying (QPSK). It's a way to convert data into a signal that can be transmitted on radio waves. Many important technologies use QPSK or its siblings:

- Direct Sequence Spread Spectrum (DSSS)
- Orthogonal Frequency Division Multiplexing (OFDM)
- Quadrature Amplitude Modulation (QAM)

Huh, you haven't heard of those either? Okay, what about these names:

**WiFi**, which uses DSSS
**3G** and **4G**, which use OFDM
**HDTV**, which uses QAM

Show of hands, how many of you use a laptop, or a tablet, or a smartphone, or a TV? I see hands from 1, 2, 3, ... everyone in the room. Congratulations, you are relying on complex numbers.

QPSK takes two bits of data and uses a table to convert it into a complex number, which is then transmitted as a phase shift (from Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$). Sending two bits isn't much, but when a radio chip does that over and over, millions of times per second, it adds up to a lot of useful stuff: email, videos, apps, cat gifs, etc. At the receiving end, a chip does it all in reverse, converting the radio wave into complex numbers, and from there back to the original data.

This rigamarole is necessary to make the signals better at handling interference. It's how so many people can use WiFi or 4G or whatever at the same time, without the signals drowning each other out. Imagine you're with a hundred people standing in the gym, holding 50 different conversations, but the person you're talking to is on the other side of the room. Complex numbers are part of the process that lets your device pick out the tiny handful of radio waves that it needs from the huge mass of noise.