To expand on Dinakar's comment about boundary value problems, the physical intuition one should have here is that the real and complex parts of a complex differentiable functions are harmonic functions (this is just a restatement of the C-R equations). An important way harmonic functions arise is as solutions of the steady-state heat equation, so one can think of the values of a harmonic function on a contour as temperatures and the values of a harmonic function in the interior of the contour as the steady-state distribution of temperature determined by the distribution of temperature on the contour. When the contour is a circle one can compute this distribution by convolving with the Poisson kernel; this is a special case of Cauchy's integral formula. The Poisson kernel itself is the canonical example of a "good kernel," and it finds application in Fourier analysis for that reason. One generally expects convolution by a good kernel to have nice properties.
In turn, one reason why diffusion should have anything to do with complex differentiability is that in both cases one wants path integrals to be homotopy invariant, in the first case because path integrals should give the difference in potential and in the second case because this is the correct generalization of the fundamental theorem of calculus.