There are some ways to define the imaginary axis.Some are obvious, like $\space Re(z)=0 \space$ others not.
I set up a condition that I think defines the imaginary axis.
Let $x$ be a real number, and $z$ a complex number.So,
$|z+x|=|z-x|$
If $\space z=x+yi$, where $x$ and $y$ are real numbers, ones get
$|x+yi+x|=|x+yi-x| \Leftrightarrow|2x+yi|=|yi| \Leftrightarrow$
$\sqrt{(2x)^2+y^2}=\sqrt{y^2} \Leftrightarrow 4x^2+y^2=y^2 \Leftrightarrow$
$4x^2=0 \Leftrightarrow x=0$
This is correct?Thanks