Perhaps the example of the calculation of fundamental groups works in this situation. Calculating, for instance, the fundamental group of the circle uses hard ideas from complex analysis (at least in many of the traditional approaches) but if one generalises to calculating the fundamental groupoid of the circle based at two points and one uses the groupoid van Kampen theorem, (not the group vKT) then the proof reduces to quite simple algebra. (The importance of that is really that it links the original complex analytic machinery into the geometric machinery in a neat way helping one to see the 'unity' or close symbiotic relationship of the two areas, which of course have a common heritage in Poincaré.)