Defining a domain for a complex number
$\mathscr{D}_w= \mathbb{C} \ \backslash \{i(x+1),i(x-1)\} \}$ is not correct. The domain should not involve the variable $x$.
$z^{2}=-i$ if and only if $z =\pm \frac {i-1} {\sqrt 2}$ so the doamin is $\mathbb C \setminus \{\frac {i-1} {\sqrt 2},-\frac {i-1} {\sqrt 2}\}$.
For the revised version of the question the domain is $\mathbb C \setminus \{-i,i\}$.