We explore oscillatory behavior in a family of periodically driven spin chains which are subject to a weak measurement followed by postselection. We discover a transition to an oscillatory phase as the strength of the measurement is increased. By mapping these spin chains to free fermion models, we find that this transition is reflected in the opening of a gap in the imaginary direction. Interestingly, we find a robust, purely real, edge π mode in the oscillatory phase. We establish a correspondence between the complex bulk spectrum and these edge modes. These oscillations are numerically found to be stable against interactions and disorder.