  The complex singlet-doublet model is a popular theory to account for dark
matter and electroweak baryogenesis, wherein the Standard Model particle
content is supplemented by a complex scalar gauge singlet, with certain
discrete symmetries imposed. The scalar potential which results thereof can
have seven different types of minima at tree-level, which may coexist for
specific choices of parameters. There is therefore the possibility that a given
minimum is not global but rather a local one, and may tunnel to a deeper
extremum, thus causing vacuum instability. This rich vacuum structure is
explained and discussed in detail.
