In addition to Qmechanic's nice answer, in string theory the wick rotation changes the Lagrangian such that the equations of motion are given by the Laplace equation instead of the conventional wave equation.

In calculations of the world sheet integral, for example for two strings to join and split again, one can make use of the conformal invariance of the Laplace equation to conformally map the integration over all world sheets to an integration over the unit disk.

In the transformed world sheet coordinates, the S-T (channel) duality can easily be seen as an equivalence of putting the injection points on the bottom and top or on the right and the left of the unit circle.