  Quantum hydrodynamics is a formulation of quantum mechanics based on the
probability density and flux (current) density of a quantum system. It can be
used to define trajectories which allow for a particle-based interpretation of
quantum mechanics, commonly known as Bohmian mechanics. However, quantum
hydrodynamics rests on the usual time-dependent formulation of quantum
mechanics where time appears as a parameter. This parameter describes the
correlation of the state of the quantum system with an external system -- a
clock -- which behaves according to classical mechanics. With the Exact
Factorization of a quantum system into a marginal and a conditional system,
quantum mechanics and hence quantum hydrodynamics can be generalized for
quantum clocks. In this article, the theory is developed and it is shown that
trajectories for the quantum system can still be defined, and that these
trajectories depend conditionally on the trajectory of the clock. Such
trajectories are not only interesting from a fundamental point of view, but
they can also find practical applications whenever a dynamics relative to an
external time parameter is composed of "fast" and "slow" degrees of freedom and
the interest is in the fast ones, while quantum effects of the slow ones (like
a branching of the wavepacket) cannot be neglected. As an illustration, time-
and clock-dependent trajectories are calculated for a model system of a
non-adiabatic dynamics, where an electron is the quantum system, a nucleus is
the quantum clock, and an external time parameter is provided, e.g. via an
interaction with a laser field that is not treated explicitly.
