  A real hypersurface in the complex quadric $Q^m=SO_{m+2}/SO_mSO_2$ is said to
be $\mathfrak A$-principal if its unit normal vector field is singular of type
$\mathfrak A$-principal everywhere. In this paper, we show that a $\mathfrak
A$-principal Hopf hypersurface in $Q^m$, $m\geq3$ is an open part of a tube
around a totally geodesic $Q^{m+1}$ in $Q^m$. We also show that such real
hypersurfaces are the only contact real hypersurfaces in $Q^m$. %, this answers
affirmatively a question posted by Berndt (cf. \cite{berndt1})}. The
classification for pseudo-Einstein real hypersurfaces in $Q^m$, $m\geq3$, is
also obtained.
