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

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.