A confusing thing about imaginary numbers happens when they are accidentally conflated with reals as if they are somehow on the same line. As we know, if we imagine imaginary numbers the same way we imagine reals, there is only a single point where the imaginary line and the real line coincide -- the origin, where both are at 0. All other points along these two lines are orthogonal, and thus never meet.
While we can describe a point at 2i, it's a point which is not on the real line nor the imaginary line, but exists somewhere in the plane that stretches between these two orthogonal worlds. Although we call this the imaginary plane, or Argand plane, etc., what is the analog for this plane in the real world?
There is no analogous "real plane", although of course the Cartesian dimensions x,y,z hint at the existence of such, and can be used that way. For example, it would be easy to imagine these three axes referring to three "realish" lines -- but we don't normally do that. There is only one real line. We only involve a plane when we're doing complex, or imaginary, math. (Note, we do use 3 planes, arranged just like the Cartesian planes, when we get to quaternions).
All this to say, when we learn that i² = -1, we're learning something that is not exactly clear, because this very simple equation is not being open about the orthogonal difference between the imaginary and real lines. We're ignoring how a point on the imaginary line can never refer to -- or "arrive at" -- a point on the real line by merely squaring i. A conflation between the two linear worlds is happening without our being clear about it, as we're doing something quite different with squaring in the imaginary world than we do with squaring in the real world. The way I see it, the conflation is happening inside the act of squaring, to be precise.
A way to sort through this is to be explicit about what i is and what its real analog, 1 is. i in expanded form is properly known as 1i (meaning 1 on the imaginary line, which is at 0 on the real line). Therefore the point at 1 on the real line can analogously be identified as 1r (meaning 1 on the real line, not 1·r). Perhaps a better nomenclature can be found, but the point is what we know as i is really 1i, and what we know as 1 is really, technically, 1r.
When we do this, we can see that i² = -1 is not using ordinary multiplication to derive a sum in the way we are used to, but is somehow talking about unseen motion in two separate dimensions. We can see this because i² -- expanded to 1i·1i -- is intuitively 1i (just as 1r·1r is obviously 1r), and actually has nothing to do with -1r on the real line.
So although "everything works" when we believe 1i·1i = -1, what we have just described is a neat hack which is not fully analyzed into an expanded form where we can see all moving pieces. There is, so to speak, a black box inside i². We are ignoring the obvious truth of how 1i and 1r are orthogonal to each other, and thus forcing the equation to say something, the truth of which is hidden from the common usage.
Note also, when we force this equation to do what it does, we are transitioning from 1 to -1 on the real line. Although it seems on the surface we're saying something like i² is like moving 1 integer in the negative direction, in fact, if you look close, we're not starting at zero. We're saying the difference between 1 and -1 on the real line, is i². This means i² is making a total movement along the real axis equal to 2, not 1.
In other words, even though the imaginary line goes through the origin at 0, we're not talking about the distance between the origin of 0 and -1, which is 1, as we might think by the shape of the equation (i² = -1) and how we commonly get to -1. Do you see how, intuitively, we're moving from horizontal positive 1 to horizontal negative 1 via a vertical i transition?
It all works, we've known that for over a century, but this is not a normal multiplication; there's a whole vertical motion which we're completely ignoring, as if some kinds of movement along the imaginary axis can be conflated with zero, having no movement at all.
If i² = -1 didn't work so well, we might have noticed this small technical detail earlier (perhaps some have), but the practical value of using i² = -1 has been immense, so great that few looked under the hood to see what was happening. I'm sure others have observed this, but I haven't encountered their work yet, so I'm open to learning what others have said in this area.
To sum: On an algebraic level, i² = -1, and that is that. On an intuitive level, though, we leap up into the imaginary domain by a quantity of 1, zoom magically over to the left by a quantity of 2, then descend back down again into the real domain by 1 again, to land upon -1. All the while, we're pretending that the leaps upward and downward didn't happen (although Euler's Identity [e^ip=-1] does describe this transition well, using an arc instead of the rectangular motion just described, the point I'm making gets even harder to understand by including e and p, so let's stay within simple orthogonal motion for now).
When the magical operation is complete, we imagine that we've remained on the real line using normal real arithmetic practices the whole time, and i² just seems like a handy tool to move quickly between the two (and also to spawn a whole new universe within math, but that's another story). It happens so quickly, we miss the prestidigitation which happened during the zoom. We miss that i² is describing a distance of 2. We miss the vertical motion. We look at the equation and imagine that i² is intuitively similar to a 1 somehow. At least, this is how I saw things for many years, until it started to unravel recently.
An underlying structural limitation of how we understand the real line is revealed by this thought experiment. I've been working on this structural issue from numerous directions for years, and all I've got so far are little articles like this and an intuitive hunch that there is a lot more going on between 1, 0, -1 (and equality, and addition, and subtraction, and multiplication, and division) than we realize.
I think there are other, related, conflations, and we really should start thinking of the "real" line as a "real sphere" which coincides more nicely with the Riemannian sphere than we realize. I am beginning to think the best intuition for numbers is spherical, not linear, and thinking they have always been intuitively spherical (you can see it quickly in the way basic counting -- and sets -- "contain" things the way that a sphere would), but we lost track of this sphere when we entered into the realm of excluded middles (led by Euclid and Aristotle).
At the very least, this insight opens the door to a whole fascinating world of what's really going on between horizontal 1 and -1 and the vertical imaginary world, so I'm delighted it all finally came together recently, and am convinced there is much more to say in this area.
original: 1:31 p.m. March 11, 2021. to be edited and illustrations added for clarity. updated and posted March 15