This is my entry into the Second Summer of Math Exposition. I used to wonder about these questions ever since my first year in university, so in a sense this is a rather personal subject for me. I hope that this video reaches someone else who is also wondering about these issues.
Here's the Desmos document for visualizing the n complex roots of any complex number:
https://www.desmos.com/calculator/1v4xdyqcos
Videos mentioned in order of apparition:
e to the pi i, a nontraditional take (old version)
https://www.youtube.com/watch?v=F_0yfvm0UoU
The 'Everything' Formula - Numberphile
https://www.youtube.com/watch?v=_s5RFgd59ao
Winding numbers and domain coloring
https://www.youtube.com/watch?v=b7FxPsqfkOY&t=245s
The 5 ways to visualize complex functions | Essence of complex analysis #3
https://www.youtube.com/watch?v=NtoIXhUgqSk
Visualizing Complex-Valued Functions
https://www.youtube.com/watch?v=1l411zv5iFA
Jeff Tupper's webpage on the graph of x^x
http://www.peda.com/grafeq/gallery/rogue/xx_exponential.html
A relevant Stack Exchange question and answer that touches on these ideas:
https://mathematica.stackexchange.com/questions/10594/how-can-i-plot-the-complex-graph-of-xx-in-mathematica
Chapters:
00:00 Intro and problem motivation
01:49 The R-to-C graph of x^a
04:37 The R-to-C graph of a^x
06:37 Rotations in complex exponentiation
07:27 The R-to-C graph of x^x
08:17 An alternative version of the graph of x^x
12:37 Recap of the various graphs of x^x
13:49 Homework
14:09 Summary and outro
Music by Vincent Rubinetti
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u