  We explore the phenomenology of a unique three-Higgs-doublet model based on
the single CP symmetry of order 4 (CP4) without any accidental symmetries. The
CP4 symmetry is imposed on the scalar potential and Yukawa interactions,
strongly shaping both sectors of the model and leading to a very characteristic
phenomenology. The scalar sector is analyzed in detail, and in the Yukawa
sector we list all possible CP4-symmetric structures which do not run into
immediate conflict with experiment, namely, do not lead to massless or
mass-degenerate quarks nor to insufficient mixing or CP-violation in the CKM
matrix. We show that the parameter space of the model, although very
constrained by CP4, is large enough to comply with the electroweak precision
data and the LHC results for the 125 GeV Higgs boson phenomenology, as well as
to perfectly reproduce all fermion masses, mixing, and CP violation. Despite
the presence of flavor changing neutral currents mediated by heavy Higgs
scalars, we find through a parameter space scan many points which accurately
reproduce the kaon CP-violating parameter $\epsilon_K$ as well as oscillation
parameters in K and $B_{(s)}$ mesons. Thus, CP4 offers a novel minimalistic
framework for building models with very few assumptions, sufficient predictive
power, and rich phenomenology yet to be explored.
