Computational Complexity studies the power and limitations of efficient computation. So is efficient computation purely an abstract mathematical object or is it trying to model a real world phenomenon? I would argue the latter. Efficient computation occurs not just in computers but in biological systems, physical systems, chemical systems, economic systems and much more. Physics focuses on the "what", computational complexity on the "how".
We don't do experiments but I do see the role of computational complexity similar to theoretical physics, trying to model the world we live in.
Pure math is the study of abstract objects, applied math develops tools used in other disciplines. Computational complexity is neither, rather an attempt to understand fundamental processes that occur in the universe, much like other sciences.