First: there seems to be a traditional belief that "pure" math fusses over tiny uninteresting details that "applied" math takes for granted, etc. Sure, we can operate this way, and make "pure math" as irrelevant as we want, or caricaturize it as such. Oppositely, if we try, we can caricaturize "applied math" as slip-shod fuzzy-thinking. :)
As I've ranted elsewhere around here, I think that the reality is not nearly as silly or binary. In particular, complex analysis is almost singular among ideas arising in 19th century mathematics, in that everything seems to work out in the best possible way. In particular, for various technical reasons that are not well-delineated by a low-tech picture of "rigor", essentially all the operations one would want to perform, ... interchange of limits, passing differentiations inside integrals, and so on, ... are legal. The problem is that the legality is not so trivial to demonstrate, and, again, low-tech demonstrations are very often semi-incomprehensible/unnatural.
In particular, the contemporary spirit of "(textbook) real analysis" runs off the road very often, apparently teaching us that we should be afraid to do anything, even the obvious. This is not constructive. The potential hazards that "real analysis" brings to our attention mostly do not occur in complex analysis... but, again, it's not so easy to prove that they don't occur... and low-tech (=undergrad, early grad-level) proofs are often extremely opaque.
I would encourage "pure" and "applied" math people alike to first approach complex analysis in a worry-free, happy-heuristic way, to see all the amazing things that it makes possible. To the extent that one worries about edge cases or technicalities, one will (but it's not so easy to see this in advance, nor by low-tech means) find that the things that might have gone wrong (in a different universe) do not do so here.
So, don't worry about "pure/rigorous" complex analysis (and certainly not about undergrad-level "real analysis", which may inhibit you more than it helps you).
(My own notes on complex analysis http://www.math.umn.edu/~garrett/m/complex/ are aimed at math grad students (without pure/applied distinction) who've not taken any course in complex analysis previously. Those notes do not really touch on "engineering" applications, but do try to illustrate how complex analysis is "the correct" fulfillment of what "calculus" should have been. Not worrying toooo much about "analysis/estimates", because it turns out there's rarely any serious issue. The notes are freely available, anyway.)