In quantum field theory, it is common to perform wick rotation $t\rightarrow -i\tau$ and get Euclidean generating functional $Z_{E}[J]$. When I first studied QFT, I just saw this a magic trick to simplify calculation. More formally (but hand-waving), I thought we just see the real time parameter $t$ extending to the whole complex plane, then we perform contour integral and identify value which is obtained from integration in real time $t$ with the value from integration in imaginary time $t$. Then we extract real parameter $\tau=it$. In the end, we have generating functional $$Z_{E}[J]=\int \mathscr{D}\phi e^{-S_{E}[\phi]/\hbar}.$$ However, I am confused whether this Euclidean generating functional is same as Minkowskian counterpart $$Z[J]=\int \mathscr{D}\phi e^{iS[\phi]/\hbar}.$$ From Cauchy theorem, they seem the same. However, $Z$ can be complex but $Z_{E}$ seems always real due to its form. A good example is the free-particle system.
One the other hand, if they are not identified, how could we use semi-classic methods such as instantons to evaluate amplitudes?
