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 (∼ 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).Simultaneous measurement of the amplitude and phase of ultrafast complex optical signals is a a key question in modern optics and photonics [1][2][3][4][5][6]. This kind of detection is needed for the characterization of various fundamental phenomena such as e.g. supercontinuum [7,8], optical rogue waves (RWs) [9-11], or soliton dynamics in mode-locked lasers [12,13]. The task remains a particulary challenging open problem when femtosecond resolution and long time windows are simultaneously required. These requirements are found for exemple in the context of nonlinear statistical optics and of the characterization of random light [14] or in the study of spatio-temporal dynamics of lasers [15].In the quest for long-window and ultrafast recording tools, temporal imaging devices, such as time-lenses, are considered as promising candidates. Time-lenses enable femtosecond time evolutions to be manipulated so that they can be magnified in time [16][17][18] or spectrally encoded [10,17,19] with high fidelity. These signals evolution replica can thus be recorded using a simple GHz oscilloscope (for time-magnification systems) or a singleshot optical spectrum analyzer [10]). No special algorithms are necessary for retrieving the ultrafast power evolutions, long windows can be recorded (up to hun-dreds of picoseconds), and the method is suitable for recording continuous-wave (i.e., non-pulsed) complex signals. Recently, temporal imaging systems have thus begun to play a central role in fundamental studies dealing with nonlinear propagation of light in fibers leading for example to the emergence of rogue waves and integrable turbulence [10,11,20], where recording long temporal traces with femtosecond resolution is mandatory. Commercial devices are also available in the market (by Pi-coLuz LLC).However, a range of applications is still hampered by the need to also record the phase evolution of long and complex ultrafast optical signals. Extension of temporal imaging has been performed in this direction, by performing heterodyning ...