Can the rule $\sqrt{xy}=\sqrt{x}\sqrt{y}$ over complex numbers be salvaged through multivaluation? Wikipedia mentions the following arithmetic fallacy: $$x<0 \land y <0 \land \sqrt{x \times y}= \sqrt{x} \times \sqrt{y}, $$ since this would lead to $-1=i^2=1$.
So, the above rule is ommitted from the rules of arithmetic of complex numbers.
Now, if we instead of thinking of arithmetic operators as functions, we consider them as relations, then would it be possible to keep the above rule?!
Let's adopt the notation $$a \ * \ b \leadsto c$$ to mean: $c$ is a result of applying operator $*$ on $a, b$.
So, $a \times b \leadsto c$ means: $c$ is a result of $a \times b$.
Then we may maintain the above rule and have: $$ i^2 \leadsto c \iff [c=-1 \lor c=1]$$

Would this lead to an inconsistency in arithmetic of complex numbers?

 A: What works is to stop worrying and learn to love that the equation $w = \sqrt{z}$, when rewritten as $w^2=z$, defines a relation on $\mathbb C$ whose graph $\{(w,z) \in \mathbb C^2 \mid w^2=z\}$ is a very interesting mathematical object to study.
A: Multiplication is a clear simple single-valued function. I wouldn't try to make it multi-valued. However, raising complex numbers to non-integer powers is inherently multi-valued, and we can salvage the identity by taking this into account.
If we define $\sqrt{\alpha}$ to be the set of all valid square roots, i.e. $\sqrt{\alpha} := \{ z \in \mathbb{C} : z^2 = \alpha\}$, then the identity works. (You need to also define the product of two sets in the obvious way: $A \cdot B = \{ab : a \in A, b \in B\}$.)
As an example, take $\alpha = -4$ and $\beta = -9$. Then we have
$$\begin{align}
\sqrt{\alpha \beta} &= \sqrt{36} = \{-6, 6\} \\
\sqrt{\alpha}\sqrt{\beta} &= \{-2, 2\} \cdot \{-3, 3\} = \{-6, 6\}
\end{align}$$
The downside of this approach is that it's just annoying to deal with sets of numbers. So understanding the multi-valued nature of $\sqrt{z}$ (or any other $z^\alpha$ with $\alpha \not \in \mathbb{Z}$) is valuable, but when writing out computations people normally just use a single square root branch and accept that this identity is no longer valid.
