  In analogy to complex function theory we introduce a Szeg\"o metric in the
context of hypercomplex function theory dealing with functions that take values
in a Clifford algebra. In particular, we are dealing with Clifford algebra
valued functions that are annihilated by the Euclidean Dirac operator in
$\mathbb{R}^{m+1}$. These are often called monogenic functions. As a
consequence of the isometry between two Hardy spaces of monogenic functions on
domains that are related to each other by a conformal map, the generalized
Szeg\"o metric turns out to have a pseudo-invariance under M\"obius
transformations. This property is crucially applied to show that the curvature
of this metric is always negative on bounded domains. Furthermore, it allows us
to establish that this metric is complete on bounded domains.
