I'm sad to see no one discussing the wave physics that this gif does a great job illustrating. Anyone who has studied waves knows that it is often useful to represent sinusoidal waves (sines and cosines) as complex exponential functions given the understanding that the real part of the exponential (which is cosine) must be used if one wants to actually measure the wave.
However, this gif illustrates an aspect of the legitimacy of the complex exponential form. A complex exponential which varies only with time (fixed space) simply rotates around a circle (like the dots in this gif). However, in the gif, if we zoom out or defocus our eyes, we can easily see planar wavefronts moving from the bottom right to the top left of the screen. In fact, these waves are sinusoidal plane waves varying in both space and time (think of it as a wave of particle density). The particle density varies along a line from the bottom right to the top left sinusoidally in space and time, but if we focus on the individual dots, we can see the beauty of the complex exponential form.
If we take the real part of these complex exponentials, the circular motion of the dots would be replaced by sinusoidal motion along the diagonal direction of the wave, and this gif would be a true representation of a compressional wave, such as sound waves.