soft question - Why non-real means only the square root of negative?

Once in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that,

There is no square root of a negative quantity, for it is not a square

However later on in 1545 an Italian mathematician Gerolamo Cardano while solving the problem $x+y=10$ and $xy=40$ obtained $x=5+\sqrt{-15}$ and $y=5-\sqrt{-15}$, although he discarded this result saying that it was "useless". But later on mathematicians like Albert Girard, Euler and W. R Hamilton introduced these complex roots with purely mathematical definition.

So this was the tale of imaginary numbers which tells us that concept of imaginary numbers was initially adopted to compensate the theory of polynomial roots (number of roots is equal to the degree of polynomial). However later on some mathematicians proposed it's practical application through geometrical interpretation and other ways.

Now I want to know that in mathematics how one can be so sure that the square root of negative numbers can be the set to be designated as non-real numbers. Or in nut shell can't there be any other definition of non-real numbers.

For example, $x^{\gamma k}=-x^{k}$ (where $\gamma$ is something similar to $\iota$ as in complex number)

or $\log{(-x)}=\gamma \log{x}$ (provided $x>0$)

Notice that I had taken the values in the functions where the input is not lying in the domain. Concept of imaginary numbers was similar to this (i.e. $\sqrt{-x}=\iota \sqrt{x}$). So in this way I'll be able to create hundreds or probably thousands of such non-real sets.

I am sorry if I am loosing some logic in this but this is more like a curiosity than a question.

Answer

If you posit $\log(-x)=\gamma \log(x)$ for all $x$, and if you want to allow the usual operations like division, you are going to be forced to conclude that $\gamma=\log(-x)/\log(x)$ for all $x$, and in particular that
$$\frac{\log(-2)}{\log(2)}=\frac{\log(-3)}{\log(3)}$$

But the various logs in this equation already have definitions, and according to those definitions, the equation in question is not true (for any of the various choices of $\log(-2)$ and $\log(-3)$.

Therefore, your $\gamma$ can exist only if you either ban division or change the definition of the log. Likewise for your other proposal $x^{\gamma k}=-x^k$.

This is why you can't just go adjoining new constants willy-nilly and declaring them to have whatever properties you want. In the case of $i$, the miracle is that you can define it in a way that does not require you to revise the existing rules of arithmetic. Such miracles are rare.