Imaginary axis of a complex analytic function has zero measure

Question

It is established in [1] that the zero set of a non-constant real analytic function on $\mathbb{R}^d$ has measure zero. To me, this result should intuitively extend to the entire imaginary axis of a non-constant complex analytic function since this is a measure zero set with respect to $\mathbb{C}$. But, I'm having a hard time showing it, particularly since the imaginary axis is an uncountably infinite set. Any suggestions here or is this not necessarily true for some obvious reason?

Answer 1

You can use the canonical identification $\mathbb C = \mathbb R^2$, which happens to preserve measure, together with the fact that the imaginary axis is the zero set of the nonconstant real analytic function $f(x,y)=x$.