Is imaginary time an extra dimension? In other words, are time and imaginary time considered two separate dimensions? If so, does imaginary time appear (as a separate dimension) in string theory (thus contributing to the 11 dimensions in M-theory)?
In physics, it is often possible/meaningful to analytically continue various real quantities into a corresponding complex variable.
In case of time, such analytical continuation allows us to wick-rotate a real time integration contour into an imaginary integration contour if we avoid singularities in the complex plane.
Wick-rotation is a mathematical procedure that doesn't count/qualify as an extra physical space-time dimension.
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.