The answer is in fact no.
A complex variety $X$ can never be a differentiable manifold (not even of class $C^1$) throughout a neighborhood of a singular point.
You can find a proof in Milnor's book "Singular Points of Complex Hypersurfaces"Hypersurfaces", Annals of Mathematics Studies 61, remark at page 13.
Notice that $X$ can be a topological manifold (i.e., a manifold of class $C^0$) in a neighborhood of a singular point. For instance, the cuspidal plane cubic $y^2z=x^3$ is homeomorphic to $\mathbb{P}^1$.
