= Solution
The continuous internal <global symmetry in field theory> is
$$
\boxed{\varphi(x)\longmapsto e^{i\alpha}\varphi(x),\qquad\alpha\in\mathbb R\text{ constant},\qquad G=U(1).}
$$
Both $|\varphi|^2$ and $\partial_\mu\varphi^*\partial^\mu\varphi$ 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 $U(1)$.
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