Following the links of Wojowu, there is a negative answer to this question for the case of self-homeomorphisms of $\mathbb C^1$, here is the answer:

Functions holomorphic on a region minus a Cantor set

So by extending the self-homeo to $\mathbb CP^1$ the answer is negative for $\mathbb CP^1$ as well. To have a positve answer, one has to require indeed that the Hausdorff dimension of $\mathbb CP^1\setminus U$ is less than $1$. (the case of $\dim=1$ seems to be still open)

Following the links of Wojowu, the answer to this question is negative for the case of self-homeomorphisms of $\mathbb C^1$, here it is:

Functions holomorphic on a region minus a Cantor set

So by extending the self-homeo to $\mathbb CP^1$ the answer is negative for $\mathbb CP^1$ as well. To have a positve answer, one has to require indeed that the Hausdorff dimension of $\mathbb CP^1\setminus U$ is less than $1$. (the case of $\dim=1$ seems to be still open)