  A system consisting of the cubic complex Ginzburg-Landau equation which is
linearly coupled to an additional linear dissipative equation, is considered.
The model was introduced earlier in the context of dual-core nonlinear optical
fibers with one active and one passive cores. We argue that it may also
possibly describe traveling-wave convection in a channel with an inner vertical
partition. By means of systematic simulations, we find new types of stable
localized excitations, which exist in the system in addition to the earlier
found stationary pulses. The new localized excitations include pulses existing
on top of a small-amplitude background (that may be regular or chaotic) {\em
above} the threshold of instability of the zero solution, and breathers into
which stationary pulses are transformed through a Hopf bifurcation below the
zero-solution instability threshold. A sharp border between the stable
stationary pulses and breathers, precluding their coexistence, is identified.
Stable bound states of two breathers with a phase shift $\pi /2$ between their
internal vibrations are found too. Above the threshold, the pulses are standing
if the small-amplitude background oscillations are regular; if the background
is chaotic, the pulses are randomly walking. With the increase of the system's
size, more randomly walking pulses are spontaneously generated. The random walk
of different pulses in a multi-pulse state is synchronized (but not completely)
due to their mutual repulsion. At a large overcriticality, the multi-pulse
state goes over into a spatiotemporal chaos.
