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?)
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.
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.)
Complex numbers and complex analysis do show up in Economic research. For example, many models imply some difference-equation in state variables such as capital, and solving these for stationary states can require complex analysis.
However, as others already emphasized, complex analysis is mostly a byproduct of solving equations. I'm not familiar with any paper where complex analysis is at the heart of the model.
Ben Tamari (1997). "Conservation and Symmetry Laws and Stabilization Programs in Economics." English.
Conservation and Symmetry Laws and Stabilization Programs in Economics Abstract: An autonomous economic system, i.e., a country, tends to be a conservative and a symmetrical system in Keynes space (Output, Money and Time [Ot, Mt ; t]), and can therefore be represented as a complex numbers system. This presentation makes it possible to aggregate (or disaggregate) the system at all levels, from the individual to the most general aggregate (and vice versa). It also offers a simultaneous solution to the problem of allocating and distributing useful resources in the market.