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Motivated by the recent CMS excess in a flavor violating Higgs decay $h\rightarrow \mu \tau$ as well as the anomaly of muon anomalous magnetic moment(muon g-2), we consider a scenario where both the excess in $h \rightarrow \mu\tau$ and the anomaly of muon g-2 are explained by the $\mu-\tau$ flavorviolation in a general two Higgs doublet model. We study various processesinvolving $\mu$ and $\tau$, and then discuss the typical predictions andconstraints in this scenario. Especially, we find that the prediction of $\tau\rightarrow \mu \gamma$ can be within the reach of the Belle II experiment. Wealso show that the lepton non-universality between $\tau \rightarrow \mu \nu\bar{\nu}$ and $\tau \rightarrow e \nu \bar{\nu}$ can be sizable, and hence theanalysis of the current Belle data and the future experimental improvementwould have an impact on this model. Besides, processes such as $\tau\rightarrow \mu l^+ l^-~(l=e,~\mu)$, $\tau \rightarrow \mu \eta$, $\mu\rightarrow e \gamma$, $\mu \rightarrow 3e$, and muon EDM can be accessible,depending on the unknown Yukawa couplings. On the other hand, the processeslike $\tau \rightarrow e \gamma$ and $\tau \rightarrow e l^+ l^-~ (l=e,~\mu)$could not be sizable to observe because of the current strong constraints onthe $e-\mu$ and $e-\tau$ flavor violations. Then we also conclude that contraryto $h \rightarrow \mu \tau$ decay mode, the lepton flavor violating Higgs bosondecay modes $h\rightarrow e \mu$ and $h\rightarrow e\tau$ are stronglysuppressed, and hence it will be difficult to observe these modes at the LHCexperiment.

$\tau$- and $\mu$-physics in a general two Higgs doublet model with $\mu-\tau$ flavor violation