  Many gauge theories possess a hidden duality between color and kinematics in
their on-shell scattering amplitudes. An open problem is to formulate an
off-shell realization of the duality, thus manifesting a kinematic algebra. We
show that 3D Chern-Simons (CS) theory in Lorenz gauge obeys off-shell
color-kinematics duality. This holds both for the gauge field and the BRST
ghosts, and the duality is manifest in the Feynman rules. A kinematic algebra
can be formulated through a second-order differential operator (Poisson
bracket) acting on the off-shell fields, and it corresponds to 3D
volume-preserving diffeomorphisms, generated by functions in Lorenz gauge. We
consider several admissible double-copy constructions of CS theory with
Yang-Mills theory, a higher-derivative (DF)^2 gauge theory, or CS theory
itself. To obtain non-vanishing amplitudes, we deform pure CS theory by
including the maximum amount of adjoint matter that respects the on-shell
duality. This gives a new formulation of an N=4 CS-matter theory, with fields
of unusual statistics. We argue that the color-stripped tree amplitudes of this
theory are equivalent to those of the Gaiotto-Witten N=4 CS theory with
bi-fundamental matter. We further show that the double copy of the N=4 CS
theory with itself corresponds to maximally supersymmetric N=8
Dirac-Born-Infeld theory.
