This feels wrong. I love it.

Imaginary numbers always feel wrong
- I never really appreciated them until watching a bunch of 3blue1brown videos. I really wish those had been available when I was still in HS.
  
  After watching a lot of Numberphile and 3B1B videos I said to myself, you know what, I'm going back to college to get a maths degree. I switched at last moment to actuarial sciences when applying, because it's looked like a good professional move and was the best decision on my life.
- After delving into quaternions, complex numbers feel simple and intuitive.
- after you spend enough time with complex numbers, the real numbers start to feel wrong
  
  Can we all at least agree that counting numbers are a joke? Sometimes they start at zero … sometimes they start at one …
- If you are comfortable with negative numbers, then you are already comfortable with the idea that a number can be tagged with an extra bit of information that represents a rotation. Complex numbers just generalize the choices available to you from 0 degrees and 180 degrees to arbitrary angles.

You need to add some disclaimer to this diagram like "not to scale"...
- It's to scale.
  Which scale is left as an exercise to the reader.
  
  I really don't think it is.

This is why a length of a vector on a complex plane is |z|=√(z×z). z is a complex conjugate of z.
- I've noticed that, if an equation calls for a number squared, they usually really mean a number multiplied by its complex conjugate.
- [ you may want to escape the characters in your comment... ]

Isn't the squaring actually multiplication by the complex conjugate when working in the complex plane? i.e., √((1 - 0 i) (1 + 0 i) + (0 - i) (0 + i)) = √(1 + - i²) = √(1 + 1) = √2. I could be totally off base here and could be confusing with something else...
- I think you're thinking of taking the absolute value squared, |z|^2 = z z*
  
  Considering we're trying to find lengths, shouldn't we be doing absolute value squared?
- Almost:
  Lengths are usually reals, and in this case the diagram suggests we can assume that A is the origin wlog (and the sides are badly drawn vectors without a direction)
  Next we convert the vectors into lengths using the abs function (root of conjugate multiplication). This gives us lengths of 1 for both.
  Finally, we can just use a Euclidean metric to get our other length √2.
  Squaring isn't multiplication by complex conjugate, that's just mapping a vector to a scalar (the complex | x | function).

It’s just dimensionally shifted. This is not only true, its truth is practical for electrical engineering purposes. Real and imaginary cartesians yay!

This is pretty much the basis behind all math around electromagnetics (and probably other areas).
- Would you explain how, for a simpleton?
  
  The short version is: we use some weird abstractions (i.e., ways of representing complex things) to do math and make sense of things.
  The longer version:
  Electromagnetic signals are how we transmit data wirelessly. Everything from radio, to wifi, to xrays, to visible light are all made up of electromagnetic signals.
  Electromagnetic waves are made up of two components: the electrical part, and the magnetic part. We model them mathematically by multiplying one part (the magnetic part, I think) by the constant i, which is defined as sqrt(-1). These are called "complex numbers", which means there is a "real" part and a "complex" (or "imaginary") part. They are often modeled as the diagram OP posted, in that they operate at "right angles" to each other, and this makes a lot of the math make sense. In reality, the way the waves propegate through the air doesn't look like that exactly, but it's how we do the math.
  It's a bit like reading a description of a place, rather than seeing a photograph. Both can give you a mental image that approximates the real thing, but the description is more "abstract" in that the words themselves (i.e., squiggles on a page) don't resemble the real thing.
  
  Circles are good at math, but what to do if you not have circle shape? Easy, redefine problem until you have numbers that look like the numbers the circle shape uses. Now we can use circle math on and solve problems about non-circles!
- Yes, relativity for example!

Now calculate the angles
- That's actually pretty easy. With CB being 0, C and B are the same point. Angle A, then, is 0, and the other two angles are undefined.
  
  A is clearly a right angle
- No thank you

Doesn't this also imply that i == 1 because CB has zero length, forcing AC and AB to be coincident? That sounds like a disproving contradiction to me.
- I think BAC is supposed to be defined as a right-angle, so that AB²+AC²=CB²
  => AB+1²=0²
  => AB = √-1
  => AB = i
  
  I mean, I see that's how they would have had to get to i, but it's not a right triangle.

Too complexe for me ;)

you are imagining things

- The reason it doesn't work is that 1 is a scalar while i is a vector (with magnitude 1). The Pythagoras theorem works with scalars, not vectors, so you'd get 1^{2 +1}2 = 2.
  
  Far as I understand it (which is not very far), i is a scalar even if you take it to be the complex number 0 + i. Just by itself i is the imaginary unit that's defined as i = sqrt(-1) (edit: or, well, the solution to x² + 1 = 0, but same difference), and nothing in that definition says it's a vector quantity.
  Even though complex numbers do extend real numbers into a 2D plane doesn't mean they're automatically vectors, and – again, as far as I've understood things – they're still treated as single entities, ie. scalars. i by itself isn't a complex number I think, though.
  The joke is that i² = -1 by definition, so i² + 1² = 0²
  Edit: eg. nothing on the imaginary number wiki page implies that the imaginary unit is not a scalar value
  
  I am sorry, but.. to be pedantic, pythagorean theorem works on real-valued length. Complex numbers can be scalars, but one does not use it for length for some reason I forgor.
- If AB = i and BC = 0, then B would be in the same 2D space as C, but one of them would be "above" the other in 3D space (which doesn't exist in this context, just as sqrt(-1) doesn't exist in the traditional sense).
  So this triangle represents a 2D object that is "standing up" on the page.
  
  It makes sense if you represent complex numbers as (a, b) pairs, where a is the real part and b is the imaginary part (just like the popular a + bi representation that can be expanded to a * (1, 0) + b * (0, 1)). AB's length is (1, 0), AC's length is (0, 1), and BC's length will also be a complex number.
  I think.
- A?= 90°