  The complex 2HDM (C2HDM) is the most general CP-violating two Higgs doublet
model that possesses a softly-broken $\mathbb{Z}_2$ symmetry. However, the
physical consequences of the model cannot depend on the basis of scalar fields
used to define it. Thus, to get a better sense of the significance of the C2HDM
parameters, we have analyzed this model by employing a basis-independent
formalism. This formalism involves transforming to the Higgs basis (which is
defined up to an arbitrary complex phase) and identifying quantities that are
invariant with respect to this phase degree of freedom. Using this method, we
have obtained the constraints that enforce the softly-broken $\mathbb{Z}_2$
symmetry. One can then relate the C2HDM parameters to basis-independent
quantities up to a two-fold ambiguity. We then show how this remaining
ambiguity is resolved. We also examine the possibility of spontaneous CP
violation when the scalar potential of the C2HDM is explicitly CP-conserving.
Basis-independent constraints are presented that govern the presence of
spontaneous CP violation.
