Đặt \(z = a + bi\,\,\left( {a,b \in \mathbb{R}} \right)\)

\( \Rightarrow \left\{ \begin{array}{l}\left| {a + bi} \right| = \left| {a + 1 + bi} \right|\\\left| {a + bi} \right| = \left| {a + \left( {1 + b} \right)i} \right|\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}{a^2} + {b^2} = {\left( {a + 1} \right)^2} + {b^2}\\{a^2} + {b^2} = {a^2} + {\left( {1 + b} \right)^2}\end{array} \right. \Leftrightarrow a = b =  - \frac{1}{2}.\)

\( \Rightarrow z =  - \frac{1}{2} - \frac{1}{2}i.\)