Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.2 Trigonometric (Polar) Form of Complex Numbers - 8.2 Exercises - Page 371: 68e

As real conjugates are symmetric across the imaginary y-axis, the Julia set must be symmetric with respect to the imaginary y-axis too.

For $z = a + bi$, real conjugate of $z$ is -a + bi. Same like complex conjugates, real conjugates also have the same absolute values. As real conjugates are symmetric across the imaginary y-axis, the Julia set must be symmetric with respect to the imaginary y-axis too.