Conjugate gauge covariant derivative ($(D_{\mu }\varphi {)}^{\ast }=({\partial }_{\mu }-ie{A}_{\mu }){\varphi }^{\ast }$)

If a complex scalar field uses $D_{\mu }\varphi =({\partial }_{\mu }+ie{A}_{\mu })\varphi$, its conjugate uses $(D_{\mu }\varphi {)}^{\ast }=({\partial }_{\mu }-ie{A}_{\mu }){\varphi }^{\ast }$. Under $\varphi \mapsto e^{-ie\alpha }\varphi$ and $A\mapsto A+d\alpha$, the two derivatives transform with opposite phases. Their product is gauge invariant. Applying the same plus-charge operator to both factors leaves an unwanted term proportional to ${\partial }_{\mu }\alpha$ and fails gauge invariance.