 Solution

The continuous internal global symmetry in field theory isBoth $|\phi {|}^2$ and ${\partial }_{\mu }{\phi }^{\ast }{\partial }^{\mu }\phi$ are unchanged. The group is the circle group, acting by phase rotations. The ordinary derivatives do not make it a local gauge symmetry: spacetime-dependent $\alpha$ would add derivative terms. There is also the discrete symmetry of complex conjugation; in the two-real-field description the full internal invariance is $O(2)$, whose connected component is this $U(1)$.