The number of complex numbers $Z$ such that $|Z - 1| = |Z + 1| = |Z - i|$ is:

$|Z - 1| = |Z + 1|$ represents the perpendicular bisector of the line joining $(1,0)$ and $(-1,0)$, i.e. the $y$–axis. For points on the $y$–axis, $Z = i y$. Now $|Z - i| = |i y - i| = |i(y - 1)| = |y - 1|$. Also $|Z + 1| = \sqrt{1 + y^2}$. So $\sqrt{1 + y^2} = |y - 1| \Rightarrow y = 0$. Thus only one point satisfies — $Z = 0$.