Answer
The graph is the vertical line $x=0$ or, equivalently, the $y$-axis.
Work Step by Step
$\begin{align*}
\vert z+1\vert &= \vert z-1\vert \\
\vert (x+iy)+1\vert &= \vert (x+iy)-1\vert \\
\vert (x+1) +iy\vert &= \vert (x-1) +iy\vert \\
\sqrt{(x+1)^2 + y^2} &= \sqrt{(x-1)^2 + y^2} \\
(x+1)^2 + y^2 &= (x-1)^2 + y^2 \\
x^2+ 2x +1 + y^2 &= x^2 -2x + 1 + y^2 \\
4x &= 0 \\
x &= 0
\end{align*}$
The graph is the vertical line $x=0$ or, equivalently, the $y$-axis.
