Motivation for Open Mapping Theorem In complex analysis, the open mapping theorem says:

A holomorphic function carries open sets to open sets.

I keep hearing that this is a deep, profound theorem. But I struggle to see why it is significant. How do I interpret it intuitively? For example, does it suggest something about how a holomorphic function locally stretches space? Does it place some sort of bound on the rate of growth of a holomorphic function near a point?
The closest I have to an intuition comes from the real analysis example,  $f(x) = x^2$, which is continuous but not open (since $f$ maps the open interval $(-1,1)$ to the interval $[0,1)$, which is not open). This happens because $f$ is not injective at $x=0$, but I'm not sure if this intuition carries over to the complex case.
I think I am misinterpreting the theorem here, so any insight would be much appreciated. Thanks!

I came across a satisfactory intuition a few days back.
A holomorphic function can never map something $2$-dimensional (like an open set) onto something $1$-dimensional (like a line or a curve). The function either maps the $2$-dimensional object to a single point (in which case, the function is constant), or it maps it to something else that's $2$ dimensional.
