Imaginary past of a quantum particle moving on imaginary time

The analytical continuation of classical equations of motion to complex times suggests that a tunneling particle spends in the barrier an imaginary duration i | T | . Does this mean that it takes a finite time to tunnel, or should tunneling be seen as an instantaneous process? It is well known that examination of the adiabatic limit in a small additional AC field points towards |T | being the time it takes to traverse the barrier. However, this is only half the story. We probe the transmitted particle's history, and find that it remembers very little of the field's past behavior, as if the transit time were close to zero. The ensuing contradiction suggests that the question is ill posed, and we explain why.