Using complex analysis to study generating functions that describe time complexity is definitely interesting, and there are some cases where using generating functions can make a complicated analysis a bit easier to deal with, but I don't see any reason to think that analytic combinatorics is more practical on a day-to-day basis than big-O notation. Analytic combinatorics is much closer in spirit to the techniques that Knuth uses to analyze performance in The Art of Computer Programming, and I don't know many programmers who prefer Knuth's style of analysis to big-O.