Does Wick rotation work for time-dependent Hamiltonian?

A Wick rotation is a method of finding a solution to a mathematical problem in Minkowski spacetime from a solution to a related problem in four dimensional Euclidean space by relating the two spaces via the substitution of the real time $t$ variable with an imaginary time $i tau$ variable.

However it requires:
The integral along the real line, $t$ real variable from $-infty to + infty$, is extended to a contour integral in the complex plane ($pm$ half plane).
The integral along the $pm$ half circle, closing the loop with an infinite radius, vanishes.
The poles, if any, of the function to be integrated are included in the original contour and remain included in the $pm pi/2$ rotated contour.
The rotated contour corresponds to the substitution $t to pm i tau$.
The contour integral is then evaluated via the residue theorem.