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You can indeed create new number systems by defining equivalence classes (quotient group).  If you take the real number line and define  x = x + 1  then you get a circle measured in turns, each time you go around a full turn you end up where you started.  This circle is still 1 dimensional.  Complex numbers are kind of special because they add a new dimension, so you get a pair of real numbers instead of just one.  You can do something similar by starting with the rational numbers and adding an irrational square root, like where a and b are rationals - multiplying or adding two numbers of this form gives a third number of the same form.  An important difference from this construction to complex numbers is that rationals extended with a square root can be mapped to the totally ordered real line, while complex numbers can't be ordered in a natural way.

https://en.wikipedia.org/wiki/Quotient_group
https://en.wikipedia.org/wiki/Field_extension

But an inequality also cannot have an answer in a natural way, right?

You're right about that. It is even solvable with simple maths. I'll have to find a better example.

as said, 2 = 0 is a totally valid result in modular arithmetic. - Also called Clock Arithmetic.

Like, on the clock, or . This sort of math is actually rather important for, for instance, cryptography, because it's rather easy to perform some tasks in this space but almost impossible to invert. If I'm not mistaken, elliptic curve protocols are essentially based on that kind of math.

What notation, the one in the wiki or the one I'm using here?
The one in the wiki gives more information and is thus clearer, so it's better in that way. However, if you explicitly say you are going to use modular arithmetic, and you specify to what you want to have it modulo, you can just use straight-up notation from normal math as always. You can also define stuff like division, exponentiation or logarithms over such spaces. Taking the logarithm is extremely hard while exponentiation is reasonable in computational complexity as far as I understood. IIRC, calculating a modular logarithm is essentially what's necessary to crack an elliptic curve encryption.

So in the below examples, when I wrote , that was in modular arithmetic, working modulo 2:
And the example on the clock: means .

It really is exactly equivalent though. Like, what you'll typically do is take a function from integers to integers and then take the result of that modulo whatever number you want to work with. However, that doesn't work as easily for inverse functions. Those need a special treatment. (That's why exponentiation is relatively easy but taking a logarithm isn't) so the inverse of this should be . Obviously, had I taken the "usual" logarithm, this would not have held. Finding this inverse in general is rather tricky to do.

These rules are not commonly used in complex numbers. Rather x² = -1 is the usual form . Assuming that was a typo ( and by god I make them all the time! ) the next part of your statement is not properly specified.

A modulo arithmetic may be what you meant originally or you may be using x in place of z, which is a conventional indicator of a complex term.

In any case keep wondering , and keep trying to express as clearly as possible what you are wondering about, because that is how you will tease out what is really of interest to you.

It's y*y=-1 not x. But if it would be x*x=1 and y*y=1 with x><y you'll get square mandelbrot set and a bitt more interesting julia sets if you use inside colouring. Like this:
http://nocache-nocookies.digitalgott.com/gallery/10/thumb_5956_21_03_12_6_28_15.jpeg
Quadratic General formula in Ultra Fractal.
If you could make general formula for y^(y-1)=-1 with x*x=1 you could see a result;)