Why don't we define "imaginary" numbers for every "impossibility"?

Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such as $x = x + 1$, or $x = 1/0$?

Edit: OK, a lot of people have said that a number $x$ such that $x = x + 1$ would break the rule that $0 \neq 1$. However, let's look at the extension from whole numbers to include negative numbers (yes, I said that I wasn't going to include this) by defining $-1$ to be the number such that $-1 + 1 = 0$. Note that this breaks the "rule" that "if $x \leq y$, then $ax \leq ay$", which was true for all $a, x, y$ before the introduction of negative numbers. So I'm not convinced that "That would break some obvious truth about all numbers" is necessarily an argument against this sort of thing.

Here's one key difference between the cases.

Suppose we add to the reals an element $i$ such that $i^2 = -1$, and then include everything else you can get from $i$ by applying addition and multiplication, while still preserving the usual rules of addition and multiplication. Expanding the reals to the complex numbers in this way does not enable us to prove new equations among the original reals that are inconsistent with previously established equations.

Suppose by contrast we add to the reals a new element $k$ postulated to be such that $k + 1 = k$ and then also add every further element you can get by applying addition and multiplication to the reals and this new element $k$. Then we have, for example, $k + 1 + 1 = k + 1$. Hence -- assuming that old and new elements together still obey the usual rules of arithmetic -- we can cheerfully subtract $k$ from each side to "prove" $2 = 1$. Ooops! Adding the postulated element $k$ enables us to prove new equations flatly inconsistent what we already know. Very bad news!

Now, we can in fact add an element like $k$ consistently if we are prepared to alter the usual rules of addition. That is to say, if we not only add new elements but also change the rules of arithmetic at the same time, then we can stay safe. This is, for example, exactly what happens when we augment the finite ordinals with infinite ordinals. We get a consistent theory at the cost e.g. of having cases such as $\omega + 1 \neq 1 + \omega$ and $1 + 1 + \omega = 1 + \omega$.

In ordinal arithmetic we have $1+\omega=\omega$. There is an algebraic downside: it turns out that $\omega+1\ne \omega$.

The short answer is that you can add any made up solution to any equation you want and extend whatever number system (or any system) you have to a larger one.

The slightly longer answer is that in mathematics it is usually with some aim in mind that an extension is made. Particularly for the imaginary numbers you mentioned, the square root of $-1$ was contemplated because it simplified manipulations on polynomials when looking for their roots.

The irrationals are added to the rational numbers since the rationals do not suffice for measuring distances (i.e., the hypotenuse of a triangle with sides equal to $1$ is $\sqrt2$).

Infinitesimals are added to the real numbers in order to make rigorous heuristic arguments using such entities.

Infinitely large natural numbers are added to the ordinary natural numbers in order to construct certain models showing the independence of certain axioms from others.

Infinite sets are added to the more tame finite sets since it is convenient to be able to talk about infinite collections of, say, numbers.

100-150 years ago 'function' assumed a very narrow meaning (not well defined) basically what we today would call: a function that is analytic everywhere except possibly at isolated points. There were even attempts to prove that every continuous function must be differentiable at almost all points. Gradually, the more exotic beasts - functions that are continuous but nowhere differentiable - entered the scene. Thus extending the study of functions from the narrow class of almost everywhere differentiable ones to the class of continuous ones. This was necessitated again by applications since such functions occur as uniform limits of analytic functions.

There are many more such examples where some extension is made fueled by some applications or a need to better understand the axiomatics of some system.

Adding an "imaginary" solution to a previous "impossible" equation always breaks existing rules (by definition, because one of the existing rules was that the impossible equation was impossible). The question is whether the gain of the new solution is worth the loss. In the case of extending reals to complex numbers, you lose the usual ordering property (an ordering $\le$ that is compatible with $+$ and $\cdot$ must have all squares nonnegative), but the resulting gain is huge because you can solve so many equations you couldn't before.

In your example of going from nonnegative numbers to all numbers, you give up the property $x \le y$ implies $ax \le ay$, but it's easy enough to fix up slightly, namely, to add the condition that $a \ge 0$ (and perhaps to say that the inequality is reversed if $a < 0$). This is also a small change.

If you add a solution to $x=x+1$, then as others have mentioned, you either have to give up $0 \ne 1$ or else give up subtraction. The first one would pretty much makes the new system useless. The second can be useful under certain circumstances. For example (as is done in measure theory, among other places), you can introduce a symbol $\infty$ that satisfies $\infty=\infty+1$. You can also define addition involving $\infty$, and most multiplications, and even most subtractions. A problem arises when you try to define the difference $\infty - \infty$, or the product $\infty \cdot 0$, so you leave those undefined. You have given up the ability to always subtract or multiply, but in some contexts that is okay. You just have to remember those restrictions when you're working in those contexts.

If you can use sets to define a structure that has some properties (like $\exists x[x=x+1]$, of course one has to know what $1$ is.), then we are done. Formal Constructions using sets is what is used to make the natural numbers, integers, rationals, real numbers,....

This part uses abstract algebra:

The ring $R[x]/\langle x^2+1\rangle$ has solutions to the equation $x^2+1=0$. (Where $1$ is the multiplicative identity of $R[x]/\langle x^2+1\rangle$)

In a similar way, the ring $R[x]/\langle1\rangle$ has solutions to the equation $x+1=x$.

This is the trivial ring. However, the trivial ring is not really interesting.