Complex analysis is used in 2 major areas in engineering - signal processing and control theory.
In signal processing, complex analysis and fourier analysis go hand in hand in the analysis of signals, and this by itself has tonnes of applications, e.g., in communication systems (your broadband, wifi, satellite communication, image/video/audio compression, signal filtering/repair/reconstruction etc). Basically, if you search for applications of signal processing, those are the applications that are indirectly the applications of complex analysis. Although most engineers will tell you that complex analysis is not necessary to "understand" signal processing, I have found that it is very helpful in going beyond simply blindly applying the fourier transform etc., to a stage where one truly understand what is going on.
See this article Connections between signal processing and complex analysis for details.
The second application area is control theory, specifically in the analysis of stability of systems and controller design. Here the word "system" is used generically, and does not necessarily refer to an electrical system. For e.g., one could use it to (try to) understand stock market movement, chemical processes/reactions. Also, control theory is used heavily in robotics, and by extension, so is complex analysis.
I should add that the complex analysis as is taught in the math department is rarely used in its "pure" form in what most people perceive as "real-world applications". For e.g., using complex analysis to help solve abstract-looking equations (e.g., differential equations) that is used to model certain interesting phenomenon (e.g., cellular processes in system biology) is also an application, although one might rarely hear people associate the two directly.
Often, engineering applications will only make use of parts of what is taught in a complex analysis course, and usually through
another area such as fourier analysis or differential equations. But this does
not mean it is "useless".