  Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean
space, we consider the maximal directional Hilbert transform $H_{\Omega}$
associated to this direction set. Our main result provides an essentially sharp
uniform bound, depending only on $N$, for the $L^2$ operator norm of
$H_{\Omega}$ in dimensions 3 and higher. The main ingredients of the proof
consist of polynomial partitioning tools from incidence geometry and an
almost-orthogonality principle for $H_{\Omega}$. The latter principle can also
be used to analyze special direction sets $\Omega$, and derive sharp $L^2$
estimates for the corresponding operator $H_{\Omega}$ that are typically
stronger than the uniform $L^2$ bound mentioned above. A number of such
examples are discussed.
