Is complex analysis used in economics? It's frequently useful in physics and engineering applications; are there any applications in theoretical economics? (If not, were there any attempts at incorporating CA that just never caught on?)

See http://en.wikipedia.org/wiki/Complex_analysis.
It should be pointed out that just because one encounters complex numbers does not mean one is doing "complex analysis", e.g. complex eigenvalues, complex Borel measures, Fourier transforms, etc. where trivial properties of complex numbers come up. 

Complex analysis is a very focused subject unlike, say, real analysis, which is eclectic by comparison. At its core are holomorphic functions of one or more complex variables. 

This paper

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=932693

is a specific instance of an economic model where complex analysis is used. The model solution technique used there is the identification between holomorphic functions on the unit disk and their continuation on the boundary. (The resulting function space is called the Hardy space, which contains the players' strategy spaces in the game being played in the paper.)