  We study the interface representation of the contact process (CP) at its
directed-percolation critical point, where the scaling properties of the
interface can be related to those of the original particle model.
Interestingly, such a behavior happens to be intrinsically anomalous and more
complex than that described by the standard Family-Vicsek dynamic scaling
Ansatz of surface kinetic roughening. We expand on a previous numerical study
by Dickman and Mu\~noz [Phys. Rev. E 62, 7632 (2000)] to fully characterize the
kinetic roughening universality class for interface dimensions $d=1, 2$, and 3.
Beyond obtaining scaling exponent values, we characterize the interface
fluctuations via their probability density function (PDF) and covariance, seen
to display universal properties which are qualitatively similar to those
recently assessed for the Kardar-Parisi-Zhang (KPZ) and other important
universality classes of kinetic roughening. Quantitatively, while for $d=1$ the
interface covariance seems to be well described by the KPZ, Airy$_1$
covariance, no such agreement occurs in terms of the fluctuation PDF nor the
scaling exponents.
