Let $X$ be a compact Haussdorf topological space with the following property:

For every continuous function $f:X\to \mathbb{C}$ the image $f(X)$ is a path connected subset of $\mathbb{C}$.

Is such a space $X$ necessarilly a path connected space?

This question is inspired by (and realated to) the following post:

Unital $C^{*}$ algebras whose all elements have path connected spectrum