How misleading is it to regard $i$ as *the* square root of $-1$? Many mathematicians regard writing $i=\sqrt{-1}$ as at best an abuse of notation, and at worst simply incorrect. This is because every non-zero complex number has two square roots, and there is no natural of way of defining the principal square root of a complex number. If $z=r\exp(i\theta)$, where $\theta\in(-\pi,\pi]$, then we could define $\sqrt{z}$ as $\sqrt{r}\exp(i\theta/2)$. However, the function $z\mapsto\sqrt{z}$ wouldn't be continuous, and it still wouldn't be true in general that $\sqrt{a}\sqrt{b}=\sqrt{ab}$. Whichever way we could go about things, defining principal square roots in the complex plane has limitations.
However, the same limitations present themselves when we have to define the complex logarithm: we still have to make an arbitrary choice about which branch we are going to use. If we want $\DeclareMathOperator{\Log}{Log}\Log z$ to be single-valued, then we have to accept that it will not be true in general that $\Log(\exp z)=z$, and nor will be it be the case that the function $z\mapsto\Log z$ is continuous. So why is it considered an abuse of notation to write $i=\sqrt{-1}$, but not an abuse of notation to write $\Log(-1)=i\pi$? Don't both of these notations obscure the fact that defining functions in the complex plane often involves making an arbitrary choice about which branch you wish to consider?
 A: It's an interesting analogy, especially since the problems both ambiguities create are related, in the equally careless sides of the "equation" $\sqrt{z}=\exp(\tfrac12\ln z)$ (one could easily multiply by $-1$ on one, $e^{\pi i}$ on the other). But it breaks down in one obvious respect. The symbol $\operatorname{Log}$ looks very different from the $\log$ we're accustomed to using on positive reals; indeed, it's not even as LaTeX-convenient (compare \operatorname{Log} with \log). Not only does this keep the "terms and conditions" of $\operatorname{Log}$ front and centre in our minds, it guarantees the symbol's introduction to students will spell out what those are.
By contrast, if you were to try to use $\log$ when you should use  $\operatorname{Log}$, every critic of "$\sqrt{-1}$" would have an entirely non-hypocritical field day. Quite a few other mathematicians would be upset too. At least those who write $\sqrt{}$ have the excuse that there's no "upper case" version of it to prevent such confusion. It's just the symbol everyone learned in primary school when taking non-negative reals' non-negative square roots.
But that gets us to the heart of why "$\sqrt{-1}$" is so dangerous. In a few characters, the reader and writer alike are forced to either work hard not to make certain mistakes, or risk making them, with they and others both likely not to notice. If someone next year invented a symbol like $\sqrt{}_{\Bbb C}$ (or something brand new I can't render in 2021), we could write down its rules, which would be as messy as the rules of $\operatorname{Log}$. But it's not been done, so let's not pretend $\sqrt{}$ is "mature" enough of a symbol to make up for it. It'll confuse students no end. Heck, it may well confuse writers no end.
Finally, it'll get even worse when you move beyond square roots, or beyond complex numbers. Can I write $i=\sqrt[4]{1}$? What about $e^{2i}=\sqrt[\pi]{1}$? In quaternions, can I write $j=\sqrt{-1}$, or $k=\sqrt{-1}$ (or uncountably infinitely many alternatives)? In split-complex numbers, can I write $j=\sqrt{1}$, or $1=\sqrt{1}$? In dual numbers, can I write $\epsilon=\sqrt{0}$, or $0=\sqrt{0}$? At this point, angels threaten to dance on our pinheads.
