I'm quite sure one could camuflage any complex analysis technique into a real calculus computation, leaving no appeal even to the notion of complex numbers. For instance, the Cauchy's formula on a circle may be translated into a formula for Fouries Fourier coefficients; the homotopy invariance of a path integral may be recovered by means of the Stokes theorem and so on. It is even possible that in some case this way one gets a simpler or more elementary computation, and sometimes, why not, it could be a reasonable and welcome operation to do (e.g. I'm thinking to the needs of an elementary course where a certain computation has to be done but the audience is supposed to have no or little Complex Analysis).
But, I'd say that in any case this would be by no means a piece of evidence against the utility of Complex Analysis techniques, nor would suggest that one can renounce to them. I think this is an a general issue about big theories. Big theories are important not only beacuse they provide useful tools to solve problems, but, even before, because they show us the way, like the Polar star. To make an example, you do not need the Open Mapping theorem of Functional Analysis to know that a certain concrete operator is invertible, if you are able to prove the convenient particular bounds, that is, just an inequality. But it's Functional Analysis who told you in advance that an inequality as the wanted one could be proved, and addressed you there.
