Imagine you're a 16th century Italian mathematician who is trying to solve cubic equations. You notice that when you try to solve some equations, you end up with a sqrt(-1) in your work. If you're Cardano, you call those terms "irreducible" and forget about them. If you're Bombelli, you realize that if you continue working at the equation while assuming sqrt(-1) is a distinct mathematical entity, you can find the real roots of cubic equations.
So I would say that it's less that "Complex numbers were invented so that we can take square roots of negative numbers", and more "Assuming that sqrt(-1) is a mathematical entity lets us solve certain cubic equations, and that's useful and interesting". Eventually, people just called sqrt(-1) "i", and then invented/discovered a lot of other math.
For small scale circuits, I like Scherz's "Practical Electronics for Inventors". For a good high level overview of the grid, along with other civil engineering topics, Grady Hillhouse's "Engineering in Plain Sight" is good. Hillhouse's videos pop up on HN occasionally (e.g., https://news.ycombinator.com/item?id=34472122)
Home automation or LEDs are always a good start in my opinion. Home automation because it's something that will affect your life and you'll notice, LEDs because they're pretty (e.g. https://news.ycombinator.com/item?id=29468919).
