  We report a detailed study of the transport coefficients of
$\beta$-Bi$_4$I$_4$ quasi-one dimensional topological insulator. Electrical
resistivity, thermoelectric power, thermal conductivity and Hall coefficient
measurements are consistent with the possible appearance of a charge density
wave order at low temperatures. Both electrons and holes contribute to the
conduction in $\beta$-Bi$_4$I$_4$ and the dominant type of charge carrier
changes with temperature as a consequence of temperature-dependent carrier
densities and mobilities. Measurements of resistivity and Seebeck coefficient
under hydrostatic pressure up to 2 GPa show a shift of the charge density wave
order to higher temperatures suggesting a strongly one-dimensional character at
ambient pressure. Surprisingly, superconductivity is induced in
$\beta$-Bi$_4$I$_4$ above 10 GPa with of 4.0 K which is slightly decreasing
upon increasing the pressure up to 20 GPa. Chemical characterisation of the
pressure-treated samples shows amorphization of $\beta$-Bi$_4$I$_4$ under
pressure and rules out decomposition into Bi and BiI$_3$ at room-temperature
conditions.
