It is not a coincidence that they arose in 16th century Italy in the context of "completing the square" and related 2D methods/intuitions for solving equations.
Yet operations on complex numbers are not the same as operations on vectors on simple two-dimensional plane. This is my point.
You can even find attempts to mix the two representations, like i+j+k vector syntax. But matrices generalize better to higher dimensions and are easier to parse.
If we take a matrix representation of a complex number it is usually done as a 2x2 matrix of very specific structure. I completely agree that it is easier to work with. But looking at them this way misses very important place of them in the grand scheme of things.
Complex numbers are actually what real numbers really ARE under the hood, we just aren't taught to think this way. 'i' is what real numbers miss to be completed. And you don't need 'j's, 'k's and others.
If your point is that introducing 'i' above traditional real numbers syntax is ugly - I completely agree.
This is an unnecessarily absolute statement. On what basis are you claiming that all number systems are fundamentally two-dimensional, and not one-dimensional, three-dimensional, or some other dimension?
I'm guessing that it is because you spent a lot of time working with mathematics in a 2D context, i.e. on paper or blackboard or screen.
I never said anything like this. I was talking about complex numbers only.
I suggest to stop here, we are talking about two different things. But, if anything, there is a comment in this thread by adrian_b which explains what I mean in more detail.
That's... not how it happened
The video (I briefly skimmed it) shows that they were invented to solve a problem in algebra. Nobody thought of them as being 2D back then.
I'm not a historian or anything, but your claim is textbook "whig history" -- and as far as I can understood you, I already proved you wrong in a previous comment.