Already in the simplest two-Higgs-doublet model with CP violation in the Hi
ggs sector, the 3 x 3 mixing matrix for the neutral Higgs bosons can substa
ntially modify their couplings, thereby endangering the "classical" Higgs s
earch strategies. However, there are sum rules relating Yukawa and Higgs-Z
couplings which ensure that the ZZ, bb and tt couplings of a given neutral
2HDM Higgs boson cannot all be simultaneously suppressed. This result impli
es that any single Higgs boson will be detectable at an e(+)e(-) collider i
f the Z + Higgs, b (b) over bar+Higgs and <(tt)over bar>Higgs production ch
annels are all kinematically accessible and if the integrated luminosity is
sufficient. We explore, as a function of Higgs mass, the luminosity requir
ed to guarantee Higgs boson detection, and find that for moderate tan beta
values the needed luminosity is unlikely to be available for all possible m
ixing scenarios. Implications of the sum rules for Higgs discovery at the T
evatron and LHC are briefly discussed. (C) 2000 Elsevier Science B.V. All r
ights reserved.