  Dynamical phase transitions are defined through non-analyticities of the
survival probability of an out-of-equilibrium time-evolving state at certain
critical times. They ensue from zeros of the corresponding survival amplitude.
By extending the time variable onto the complex domain, we formulate the
complex-time survival amplitude. The complex zeros of this quantity near the
time axis correspond, in the infinite-size limit, to non-analytical points
where the survival probability abruptly vanishes. Our results are numerically
exemplified in the fully-connected transverse-field Ising model, which displays
a symmetry-broken phase delimited by an excited-state quantum phase transition.
A detailed study of the behavior of the complex-time survival amplitude when
the characteristics of the out-of-equilibrium protocol changes is presented.
The influence of the excited-state quantum phase transition is also put into
context.
