The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i|$ equals:

$|z-1|=|z+1|$ represents the perpendicular bisector of the segment joining $(1,0)$ and $(-1,0)$, which is the $y$ -axis ($x=0$).$|z+1|=|z-i|$ represents the perpendicular bisector of the segment joining $(-1,0)$ and $(0,1)$.The intersection of two non-parallel lines in the plane is exactly one point.That point is the circumcenter of the triangle with vertices $(1,0),(-1,0),(0,1)$, which is the origin $(0,0)$.Thus, only one such complex number exists ($z=0$).