How misleading is it to regard $i$ as *the* square root of $-1$?

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.