Well the replacement of time by imaginary time makes the transition from QFT to statistical mechanics and the other way around. I mean, we know it works but we do not know if there might be some underlying system that governs this behaviour...I mean, this is a bit like working with complex numbers : in order to describe something real in an easy way, it is favourable to 'make a transition to un-real numbers' and do the math, the results are conform the observed reality. If you come to think about this, it is very counter-intuitive.
Besides, keep in mind that this WIck contraction (ie analitical continuation of the gaussian integral in the path-integral formalism) works practically everywhere in QFT !
If I recall, the analytic continuation is to a cyclic imaginary time coordinate rather than just an analytic continuation. The reason is basically that QM and statistical mechanics are both probability theories where the probability is given by an exponential. The statistical mechanical exponent is real (energy multiplied by [tex]-k/T[/tex]), while the QM exponent is imaginary (energy multiplied by [tex]i\hbar[/tex]). The exponential causes the time part of the integral to end up cyclic.
My speculation for the coincidence is that it is the statistical mechanics that is fundamental and quantum mechanics is the accidental equality. To do this, you have to have time cyclic and imaginary, which tells us how "proper time" can be treated as a coordinate.