  The traditional adjacency matrix of a mixed graph is not symmetric in
general, hence its eigenvalues may be not real. To overcome this obstacle,
several authors have recently defined and studied various Hermitian adjacency
matrices of digraphs or mixed graphs. The usual idea is to choose a complex
number $\alpha \in \mathbb{C}$ with $\vert \alpha\vert=1$. The traditional
adjacency matrix is modified such that the value $\alpha$ (resp.\ $\bar\alpha$)
replaces the previous value at position $(i,j)$ whenever the only edge between
the vertices $i$ and $j$ is an arc from $i$ to $j$ (resp.\ from $j$ to $i$). So
far, the choices $\alpha=i$ and $\alpha=e^{\frac{\pi}{3}i}$ have attracted
particular interest. It has even been argued that the latter choice might be
ideal, in a certain sense. The present paper extends and generalizes previous
results and, moreover, proposes another particular choice for $\alpha$ that
appears just as favourable, if not more.
