  We consider a conformal complex singlet extension of the Standard Model with
a Higgs portal interaction. The global $U(1)$ symmetry of the complex singlet
can be either broken or unbroken and we study each scenario. In the unbroken
case, the global $U(1)$ symmetry protects the complex singlet from decaying,
leading to an ideal cold dark matter candidate with approximately 100 GeV mass
along with a significant proportion of thermal relic dark matter abundance. In
the broken case, we have developed a renormalization-scale optimization
technique to significantly narrow the parameter space and in some situations,
provide unique predictions for all the model's couplings and masses. We have
found there exists a second Higgs boson with a mass of approximately
$550\,\rm{GeV}$ that mixes with the known $125\,\rm{GeV}$ Higgs with a large
mixing angle $\sin\theta\approx 0.47$ consistent with current experimental
limits. The imaginary part of the complex singlet in the broken case could
provide axion dark matter for a wide range of models. Upon including
interactions of the complex scalar with an additional vector-like fermion, we
explore the possibility of a diphoton excess in both the unbroken and the
broken cases. In the unbroken case, the model can provide a natural explanation
for diphoton excess if extra terms are introduced providing extra contributions
to the singlet mass. In the broken case, we find a set of coupling solutions
that yield a second Higgs boson of mass $720\,\rm{GeV}$ and an $830\,\rm{GeV}$
extra vector-like fermion $F$, which is able to address the $750\,\rm{GeV}$ LHC
diphoton excess. We also provide criteria to determine the symmetry breaking
pattern in both the Higgs and hidden sectors.
