In the Standard Model, the Higgs field has two complex components. The "two" comes from how it couples to the $SU(2)_L$ gauge field, the one that couples only to the left-handed components of the fermions. Even without the $SU(2)_L$ gauge field, the "complex" would still be related to the fact that it also couples to the $U(1)_Y$ gauge field, the one whose charges are called "hypercharges." Under a $U(1)_Y$ gauge transformation $A_\mu\rightarrow A_\mu+\partial_\mu\theta$, the Higgs field transforms as $\Phi\rightarrow \exp(i\theta)\Phi$. As noted in the OP, we could think of it as four real fields instead of two complex fields, but the complex representation is easier to manage.
More generally, any time the scalar field is "charged" with respect to a $U(1)$ gauge field, representing it as a complex field tends to be easier to manage than representing it as a pair of real fields. For example, the Abelian Higgs model (which is sometimes studied in the context of superconductivity, for example) involves a complex scalar field.