Several motivating physical applications are listed on wikipedia
Why do we need to study numbers which do not belong to the real world?
You may want to stoke the students' imagination by disseminating the deeper truth - that the world is neither real, complex nor p-adic (these are just completions of Q). Here is a nice quote by Yuri Manin picked from here
On the fundamental level our world is neither real nor p-adic; it is adelic. For some reasons, reflecting the physical nature of our kind of living matter (e.g. the fact that we are built of massive particles), we tend to project the adelic picture onto its real side. We can equally well spiritually project it upon its non-Archimediean side and calculate most important things arithmetically. The relations between "real" and "arithmetical" pictures of the world is that of complementarity, like the relation between conjugate observables in quantum mechanics. (Y. Manin, in Conformal Invariance and String Theory, (Academic Press, 1989) 293-303 )
