  We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at
least one singular branch are equivalent by a (real) smooth diffeomorphism,
then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result
was shown by Ephraim for irreducible hypersurfaces before, but his proof is not
constructive. Indeed, we show that the complex isomorphism is given by the
Taylor series of the diffeomorphism. We also prove an analogous result for the
case of non-irreducible hypersurfaces containing an irreducible component of
zero-dimensional isosingular locus. Moreover, we provide a general overview of
the different classifications of plane curve singularities.
