*ignore what I wrote below. I had pythag in my head with the diagonal line thing. I suppose you could substitute counting squares down off the paper onto the table instead of cutting across imaginary diagonals… but I’m tired and don’t feel like rewriting this brain fart.
Read ahead with caution to a tired brain 
ORIGINAL answer.. which is unbelievable wrong.:
Here’s the blame:
The whole “i” thing has a lot of spook and awe around it.
But look: What’s a square root?
You’re making a square of a piece of paper let’s say. You want to see the distance from corner to corner.
But there’s a problem: You can’t use square units to make a diagonal.
Ah, but look: Square root. Problem solved. You fold the paper down the diagonal and you measure it. Done.
But you’re left with an annoying bit. That diagonal doesn’t exist in nice squares. Thus begins the issues. This issue shows up in analog to digital _all_ of the time. Want square pixels to make a diagonal line? You can’t. You can FAKE IT but at some point, you gotta face the gaps.
So ok. negative numbers. They have some clear rules in arith. Add, subtract, multiple, divide. Simple enough anyway.
But remember, when you’re dealing with numbers, they’re on a one dimensional line. You want two dimensions, you go to a 90 degree angle from zero and go up and down. Same number line, running 90 degrees up.
You got lots of tricks you can do there, but they’re always running at 90 degrees of each other, and you’re playing with the spaces inbetween.
But when you want to “cut across” that nice grid, you gotta play with geometry tricks.
These start getting weird quickly.
Want that perfect circle? Eh, you ain’t gonna really get it. You can’t entirely turn circles into squares, just get close-ish.
But, generally good enough. Like the square root, folding those squares across the diagonal and you’re generally fine.
But then you get into real issues. You start leaving the paper!
-1. Nice -1 sits there on the one dimensional number line.
You can slide it back and forth. You can move it higher and lower. All sorts of neat things with the -1.
Then you say, “Hey, I want to make a diagonal connecting -1 to -1 on a grid.
GREAT! Just … wait. hang on….
-1 is off the paper. It’s on the table somewhere. Whatever. You got a negative 1 sitting on the left side of the paper on the table.
You got another negative 1 sitting just below the paper on the same table.
So ok, let’s FOLD THE PAPER across the diagonal in a place where there IS NO PAPER.
Well, maybe we can do some stuff with circles and get PART of that -1 to -1 diagonal (the square root) back ONTO the paper.
Oh good! We can do that. Some of it. Some nice geometry tricks gets some lines going back onto the paper.
Just not all of it. But enough to get SOME real number action there. An arc of a circle here and there. Ok. Good ol’ Pi to the rescue.
Now you’re like, “Hm, this imaginary spot off the paper that i can get partly onto the paper but not entirely… I’m going to take the diagonal line that I can’t draw anywhere – the square root of -1… and not go up a 90 degree angle into a square… but a dimensionality that’s the same as itself.
Eeerrrr.. nothing’s on the paper here.
Well, some of it is. Enough of it is to work _something_ out
and one of the answers ends up on the paper. 0.2ish
Am I explaining this entirely the wrong way? ABSOLUTELY. Would this be a terrible way to teach math? yes it would.
But it’s how I happen to see numbers and how I can mentally justify i^i ending up on the paper.
I mean, you’re dealing with Pi in there for example. You’re dealing with a 2. You got some real number stuff happening here. Stuff that’s on the number line. Stuff that’s on paper.
That we can do some flipping to get an imaginary, not fully measurable ‘something’ back onto the paper from somewhere in an impossible to measure vicinity shouldn’t be surprising.
Anyway, it’s how I see math. I flip things around, abstract numbers to the number line and things like square roots I see as folding paper diagonally…and stuff.
It’s a naive view of this stuff but it’s how I reconcile it somewhat. Even if it’s wrong, I’m ok with that.
It’s salvaged somewhat by keeping the concept of tying the number line to a piece of paper. and other ways where you try to remember what analogues these mathematical functions have when applied to paper sitting on a tabletop.