The square root of a variable is negative? If the square root of a variable is negative, as shown below:

$$\sqrt x = -1$$

Then what is $x$ equal to? The closest answer I can think of is $i^4$.

$$\sqrt{i^4}=i^{\frac42}=i^2=-1$$

But if $i^4$ is evaluated first, then it doesn't work:

$$\sqrt{i^4}=\sqrt{(-1)(-1)}=\sqrt1=1$$
If a negative number is a square root of $x$, then $x$ is the square of that negative number, and thus $x$ is a positive number.  Every nonzero complex number has two square roots.  However, when you talk about the square root of a positive number, you generally mean the positive square root, not the negative square root.
Using the Quaternion definition, $ x $ can be any of $i^2j^2$, $j^2k^2$ or $i^2k^2$, without using any arbitrairy operations order.

If $\sqrt{x} = -1$, $ x = -1 * -1$ and $-1$ can be defined by $i^2$, $j^2$ and $k^2$