  Using certain solutions of the curve shortening flow, including
self-shrinking and self-expanding curves or spirals, we construct and
characterize many new examples of translating solitons for mean curvature flow
in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones
\cite{JLT} in dimension two. The simplest (non trivial) example in our family
is characterized as the only (non totally geodesic) Hamiltonian stationary
Lagrangian translating soliton for mean curvature flow in complex Euclidean
plane.
