  We present new imaging and spectroscopic observations of the exoplanet host
star $\gamma$ Cep A, and of its low-mass stellar companion $\gamma$ Cep B. We
used AstraLux/CAHA to follow the orbital motion of the companion around its
primary, whose radial velocity was determined with spectra of the star, taken
with the spectrograph FLECHAS at the University Observatory Jena. We measured
the astrometry of $\gamma$ Cep B relative to its primary in all AstraLux images
and determined its apparent SDSS i'-band photometry, for which we obtained
i'=9.84$\pm$0.17 mag. Using stellar evolutionary models and the Gaia parallax
of the exoplanet host star, we derived the mass of $\gamma$ Cep B to be
0.39$\pm$0.03 M${_\odot}$. This is in good agreement with the mass of the
companion, derived from its NIR photometry, given in the literature. With the
detection limit, reached in our AstraLux images, we explored the detection
space of potential additional companions in the $\gamma$ Cep binary system. In
the background limited region at angular separations larger then 5 '' (or 69 au
of proj. separation) companions down to 0.11 M$_\odot$ are detectable around
$\gamma$ Cep A. The radial FoV, fully covered in our AstraLux images, exhibits
a radius of 11.2 ''. This allows the detection of companions with proj.
separations up to 155 au. However, except for $\gamma$ Cep B no additional
companions could be imaged around the exoplanet host star. We redetermined the
orbital solution of the $\gamma$ Cep binary system with the new AstraLux
astrometry of $\gamma$ Cep B and the additional radial velocities of $\gamma$
Cep A, obtained from our FLECHAS spectroscopy of the star, combined with
astrometric and radial velocity data from the literature. The determined
orbital elements were used to derive the system parameters and to calculate
specific future ephemeris for this intriguing exoplanet host binary star
system.
