[Coda]

A complex number is a real number (which might be 0, so all imaginary numbers are complex numbers) plus an imaginary number (which might be 0i, so all real numbers are complex numbers).

Yes, norm = magnitude = absolute value.

Imaginary numbers are put on the y axis by convention purely because the real number line is usually drawn horizontally. It doesn't actually matter, but the convention makes it possible for people to look at a graph together and agree on what's being plotted.

Real numbers are one dimensional because they're on a number line? And then having two number lines as the x and y axis they're...plotted as two dimensional??
Also, if real numbers are one dimensional and are plotted in two dimensions, then imaginary numbers should be one dimensional and plotted in two dimensions too! Maybe..?

Yes, if you look at JUST imaginary numbers, then they are indeed one-dimensional. However, the main reason you can't plot a function using imaginary numbers in two dimensions is because if you multiply two imaginary numbers together, you get a real number, so you wouldn't have a way to plot it. You COULD meaningfully plot it in THREE dimensions, and you'd get a single curve, not a surface.

Complex numbers are a combo of real numbers and imaginary numbers put together. How that works, I'm guessing, is by having a figure with four axis, two for imaginary and two for reals and somehow one point can be made from that? But how that works gets into the difficult visualization part.

Yes, you're completely right. This is sometimes graphed using color to represent some of the axes (often hue for one of them and brightness for the other), or it can be roughly sketched using the picture like I showed above.

It's possible to think of it as a "field." What is a field..?

You're familiar with electric fields, magnetic fields, and gravitational fields already. You can correspond every point within those fields to some useful value, such as "the force an electron would feel at this point". The same is true here: You can correspond every point on the xy plane with the value of the function at that complex number, and draw some representative examples, like in the picture I showed.

In fact, the example I showed could be interpreted as a magnetic field, with a bar magnet laid upon the x axis. You can see the vectors tracing out curves that go from one pole of the magnet to the other. There's an actual physical realization of this -- if you put a sheet of paper over such a magnet, and sprinkle iron filings over the paper, they will actually align themselves with that picture!

From this point forward I'm confused.
What is f(x + yi) and what is the vector referring to?

Just like you're used to seeing f(x) referring to a function over the real numbers, f(x + yi) is referring to a function over the complex numbers. The vector refers to the value of the function at that point -- a vector pointing straight right might represent 1+0i, one pointing straight up might represent 0+1i, one pointing up-right (and a little longer than the other two) might represent 1+1i, et cetera. The length of the vector corresponds to the norm (= magnitude = absolute value) of that complex number.

In the graph picture, I guess the x-axis turned into an axis that for complex numbers because of the problem with how difficult it is to visualize three dimensions? But that can't be it, surely? I'm definitely guessing here. And besides, why have just one axis take on complex values? Why not both? Is that even possible?

No, one axis isn't representing complex numbers. It does indeed require BOTH axes. The x axis is the real part of the input to the function, and the y axis is the imaginary part. That's why it's written f(x + yi).

As for the last bit I still need to follow it on pen and paper, but essentially it's possible for something to just have complex roots!

Yep, that's correct. They can still be called imaginary roots, from the perspective of them not being real roots, but complex roots is technically more precise.

Not math question: What is a "pacifying issue"? I read somewhere that someone thought that, for sure, the government was thinking (1970) that Earth Day or something environmental was a "pacifying issue." Does it mean an issue the government wants to just ignore..? Thinks will just die after a while?

It's an issue that is brought up to attract focus to something that they think will bring peace to the public. Earth Day wasn't there for the purposes of ACTUALLY protecting the environment. It was created to make people feel good about thinking about the environment.

By calling it a "pacifying issue" the implication is that it's distracting away from more controversial issues.