Applications of Signed/Complex Measures While first studying measure theory, I found signed and complex measures to be quite exciting; I recognized complex analysis as a powerful tool, and figured complex measures would be important as well. Now, after three courses in measure theory, I find myself realizing I have never once seen an application of either signed or complex measures (beyond perhaps the space of signed measures naturally being a vector space, which can allow some additional proof techniques for the Radon-Nikodym theorem).
One reason for this is obvious:

Hahn-Jordan Decomposition: Given any complex measure, we can decompose it into its real and imaginary parts and then (canonically)
decompose it into a linear combination of four non-negative measures.

Using this theorem as justification, every source I have found treats non-negative measures as the fundamental concept, and relegates signed and complex measures to the wayside, except perhaps for a complex version of Radon-Nikodym. This has never sat well with me, as I would think there should be contexts in which signed (or, more generally, complex) measures should arise naturally, in which case decomposing the measure could be detrimental. With this in mind, I would ask the following related questions:

Question 1: What contexts, if any, are there in which signed or complex measures arise naturally?
Question 2: Regardless of naturality, have signed/complex measures found any nontrivial uses beyond first decomposing into a sum of non-negative measures?

I recognize these questions are rather open-ended and would thus accept any insight, be it large or small.
 A: If $f$ is a non-negative measurable function on $(X,S,\mu)$ then $\nu(E)=\int_E fd\mu$ defines a positive measure $\nu$. But we often have to deal with this set function $\nu$ when $f$ is just integrable, not necessarily non-negative. Real valued $f$ gives a real measure and complex valued $f$ gives a complex measure.
One of the most important applications of complex measures is found in Riesz Representation Theorem. A linear map $T$ on the space of continuous functions on $[0,1]$ (or more generally a compact Hausdorff space) with the sup norm is continuous iff we  can write $Tf=\int f d\mu$ for some regular Borel complex measure $\mu$.
A: Basically the main reason people care about complex measures is Fourier/complex analysis. Discrete Fourier and similar transforms are important in applied mathematics (image compression, speech analysis, electrical engineering, etc...) and this lends to calculus with complex valued functions (and functions can often be thought of as measures through duality, via R.R.T.) In short, measure theory is a tool to better understand and formalize calculus concepts and complex numbers arise naturally there.
One can of course avoid complex measures altogether by just breaking measures in real and complex parts, but it is just more natural (the notation is more intuitive for instance) to work with complex measures. This is why for ages people have written contour integrals as they do.
