This is a question about problem 3 from Ahlfors section 3.1, which states
Prove that the most general transformation which leaves the origin fixed and preserves all distances is either a rotation or a rotation followed by reflexion in the real axis.
Now it has been answered here before where they show that if $f$ is indeed a fractional linear transformation then $f(z)=az$ where $|a|=1$. But when it's not, they somehow conclude the only two options are $f(z)=az$ or $a\bar{z}$(here the conjugate is what you get after reflecting in the real axis). Firstly, Ahlfors doesn't even define what "general transformations" are, but assuming they are perhaps continuous/holomorphic maps $\mathbb{C}\to \Bbb{C}$. I don't see why it has to be only one of these two kinds in such a scenario. So I guess my question is :
If $f:\mathbb{C}\to \Bbb{C}$ a continuous map preserving the origin and preserving lengths, then $f(z)$ or $f(\bar{z})$ is a linear fractional transformation. Then since for linear fractional transformations we know the answer from the linked answer, the full claim follows.
Is this a true claim? Is there any way to prove it? I think as stated the problem doesn't even allow $f$ to have any kind of structure, linear or holomorphic etc. So I don't see why we should even consider fractional transformations to begin with. I feel like this shouldn't be as difficult of a problem if only Ahlfors defined the terminology clearly and this is making me lose my mind. Any help would be appreciated.