Reflections on the q-Fourier transform and the q-Gaussian function

We appeal to a complex q-Fourier transform as a generalization of the (real) one analyzed in [Milan J. Math. {\bf 76} (2008) 307]. By recourse to tempered ultra-distributions we are able to show that the q-Gaussian distribution can be nicely obtained, overcoming all troubles that afflict its real counterpart.