Tomorrow, 60 Minutes will air a segment called, no joke, Elephant Language (HT Daily Beast). It's about a group out of Cornell called the Elephant Listening Project. who believe that the low-frequency infrasonic sounds made by elephants might constitute a language. I am naturally suspicious because these kinds of claims tend to conflate the notion of language with the more general notion of communication system into a muddled mess of a concept. Without a good definition of human language, how can we say that some non-human communication system is also a "language." It's an untestable claim.
There are thousands of human language problems to solve, and few linguists to solve them. Investigating elephant language is low on the priority list, I'd say. As I've noted here, animal language stories are just one of those things that gets regular people to say, "gee wiz, really? wow" while it gets academic linguists to say "meh."