Answer
The graph is the line $x+y=0.$
Work Step by Step
$\begin{align*}
\vert z+i\vert &= \vert z-1\vert \\
\vert (x+iy)+i\vert &= \vert (x+iy)-1\vert \\
\vert x +i(y+1)\vert &= \vert (x-1) +iy\vert \\
\sqrt{x^2 + (y+1)^2} &= \sqrt{(x-1)^2 + y^2} \\
x^2 + (y+1)^2 &= (x-1)^2 + y^2 \\
x^2+ y^2 + 2y + 1 &= x^2 -2x + 1 + y^2 \\
2y &= -2x \\
2x+2y &=0 \\
x+y &=0
\end{align*}$
The graph is the line $x+y=0.$
