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Does Shelah's categoricity conjecture for abstract elementary classes have applications in other branches of mathematics?

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I’m not sure if it counts as “applications in other branches of mathematics”.

The other branch could be category theory. The conjecture was translated in the category-theoretic framework:

Beke, Tibor, and Jirí Rosický. "Abstract elementary classes and accessible categories." Annals of Pure and Applied Logic (2012) (arxiv link)

The above work can be seen as a part of research in categorical model theory which was also mentioned on the MO before (see here and here).

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Shelah's catgegoricity conjecture has no applications to other fields. It was proposed as a test problem to measure maturity and power of current devices and concepts. In that role it served very well (it is still open). It is interesting to point out that Morley categoricity theorem even 50 years after the proof still has no real applications, however teh concepts developed in the proof are of central value in applications. –  Rami Grossberg Mar 9 at 15:59

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