  We chose the $\mu$-term extended next-to-minimal supersymmetric standard
model ($\mu$NMSSM) for this work, and the phenomenological research is based on
the assumption of double Higgs resonance state as the Standard Model (SM)-like
Higgs considering the recent $(g-2)_\mu$ result. The study also take into
account a variety of experimental results, including direct detection of dark
matter (DM) and searching results for sparticles at the Large Hadron Collider
(LHC). We study the characteristic of DM confronted with limitations of direct
detection experiments. Following that, we concentrate on the properties of the
mass-degenerate SM-like Higgs bosons and explaining the anomaly of $(g-2)_\mu$.
We conclude that the anomaly of $(g-2)_\mu$ can be explained in the scenario
with two mass-degenerate SM-like Higgs, and there are samples that meet all
current constraints and outperform SM in fitting Higgs data.
