  Temporal imaging systems are outstanding tools for single-shot observation of
optical signals that have irregular and ultrafast dynamics. They allow long
time windows to be recorded with femtosecond resolution, and do not rely on
complex algorithms. However, simultaneous recording of amplitude and phase
remains an open challenge for these systems. Here we present a new heterodyne
time-lens arrangement that efficiently records both the amplitude and phase of
complex signals, while keeping the performances of classical time-lens systems
($\sim 200$~fs) and field of view (tens of ps). Phase and time are encoded onto
the two spatial dimensions of a camera. We demonstrate direct application of
our heterodyne time lens to turbulent-like optical fields and optical rogue
waves generated from nonlinear propagation of partially coherent waves inside
optical fibres. We also show how this phase-sensitive time-lens system enables
digital temporal holography to be performed with even higher temporal
resolution (80 fs).
