Is it true that "there is no such thing as the square root of minus one"? Is the statement "there is no such thing as the square root of minus one" a true statement?
It seems to me that we need to be careful about the word "the" as it appears in the statement. If we see it as implying uniqueness, then surely the statement is true after all, since $i$ and $-i$ are distinct square roots of minus one.

Indeed, as suggested in the question, if we were to speak of the root of $-1$, then there should be just one such value, or at least one of the two possible values should be distinguished.
For positive reals, we can speak of, for instance, the square root of $4$, because out of the two possible values of $+2$ and $-2$, only one is positive (and this is the one we take). Note that numbers $+2$ and $-2$ are quite different: for instance, $+2$ is a square of a real number, and $-2$ is not. 
For $-1$, things are a little trickier. There are two possible choices of the root, $+i$ or $-i$. And in some sense, these choices are both equally legitimate. There is no way to distinguish $+i$ from $-i$ using only arithmetic operations and knowing what the real numbers are. This is a consequence of the fact that the conjugation mat $z \mapsto \bar{z}$ preserves multiplication, addition, and leaves $\mathbb{R}$ invariant.
