There's a deeper concept behind this fake proof Whenever I start to get cocky about my math knowledge, some basic property of the complex plane puts me back in my place. I just came across this fake proof that 2 = 0:

In the shallowest sense, I think I know what the problem is here. The author deceptively exploits how squaring is a 2-to-1 mapping on $C \setminus 0$, and flips the sign of a root somewhere. If we use $-i$ as the square root of $-1$ instead of $i$, we get a tautology rather than a contradiction.
But what I'd like is a fuller description of the phenomenon underlying the trick. My guess is that the proof smuggles in some special property of $R$ that is so basic that the naive reader assumes it holds in $C$, even though it doesn't. Could anyone state what this property is? Is my guess on the right track?

The proof tacitly assumes that there is a function $sqrt:\mathbb{C}\rightarrow\mathbb{C}$ (which it calls "$\sqrt{\cdot}$") with the following two properties: