  We extend heavy Higgs searches at the Large Hadron Collider for $H \to \tau
\mu$ by CMS, and $H \to \tau\tau$ by ATLAS and CMS, to study discovery
prospects of extra Higgs states in $pp \to H,A \to \tau \mu, \tau\tau$ with
$e\mu + E_T$ and $j_{\tau} \mu + E_T$ final states, where $j_{\tau} = \pi \, ,
\rho\, , a_1$ and $E_T$ is missing transverse energy. In a general two Higgs
doublet model without $Z_2$ symmetry, extra Yukawa couplings $\rho_{\tau\tau}$
and $\rho_{\tau\mu}$ can drive $H,A \to \tau\tau$ and $\tau\mu$ channels at
hadron colliders, following gluon-gluon fusion production with extra
$\rho_{tt}$ couplings. The light Higgs boson $h(125)$ is found to resemble
closely the Standard Model Higgs boson; in the alignment limit of $\cos\gamma
\to 0$ for $h$--$H$ mixing, flavor changing neutral Higgs couplings such as $h
\to \tau\mu$ are naturally suppressed, but the couplings of the heavier $H$ is
optimized by $\sin \gamma \to 1$. We define various signal regions for $H,A \to
\tau\mu$ and $\tau \tau$ and evaluate physics backgrounds from dominant
processes with realistic acceptance cuts and tagging efficiencies. Two
different studies are presented. We first perform a parton level study without
any hadronization and with minimal detector smearing. We then include
hadronization using PYTHIA 8.2 and fast detector simulation using DELPHES to
give event level simulation. Results for $\sqrt{s}= 13$ TeV appear promising,
which we extend further to $\sqrt{s} = 14$ TeV for the High Luminosity LHC.
