### Instruction: Find the complex number $z$ such that
\[|z - 1| = |z + 3| = |z - i|.\].### Response: This problem involves the modulus or absolute value of complex numbers, which is the distance from the origin in the complex plane.
The equation $|z - 1| = |z + 3|$ means that $z$ is equidistant from $1$ and $-3$ on the real axis.
This implies that $z$ lies on the perpendicular bisector of the segment joining $1$ and $-3$, which is the line $x = -1$.
Similarly, the equation $|z - i| = |z - 1|$ means that $z$ is equidistant from $i$ and $1$ on the imaginary axis.
This implies that $z$ lies on the perpendicular bisector of the segment joining $i$ and $1$, which is the line $y = \frac{1}{2}$.
Therefore, $z$ must be the intersection of the two lines $x = -1$ and $y = \frac{1}{2}$.