  We reexamine the Parisi-Klauder conjecture for complex e^{i\theta/2} \phi^4
measures with a Wick rotation angle 0 <= \theta/2 < \pi/2 interpolating between
Euclidean and Lorentzian signature. Our main result is that the asymptotics for
short stochastic times t encapsulates information also about the equilibrium
aspects. The moments evaluated with the complex measure and with the real
measure defined by the stochastic Langevin equation have the same t -> 0
asymptotic expansion which is shown to be Borel summable. The Borel transform
correctly reproduces the time dependent moments of the complex measure for all
t, including their t -> infinity equilibrium values. On the other hand the
results of a direct numerical simulation of the Langevin moments are found to
disagree from the `correct' result for t larger than a finite t_c. The
breakdown time t_c increases powerlike for decreasing strength of the noise's
imaginary part but cannot be excluded to be finite for purely real noise. To
ascertain the discrepancy we also compute the real equilibrium distribution for
complex noise explicitly and verify that its moments differ from those obtained
with the complex measure.
