It could mean that the dimensionless constant "alpha", also known as the "fine structure constant", varies, and the simplest way to express that change mathematically is as a change in C rather than in one of the other parameters (such as the charge of the electron).
(I read a book about this.)
New Salt Compounds Challenge the Foundation of Chemistry
The title is overblown (it depends on what you think the foundation is), but get a load of this:
And here's the philosophical bit:
The obvious example of local truth is relativistic effects being pretty much invisible over the durations and distances that are normal for people, but there's also that the surface of the earth is near enough to flat for many human purposes.
Any suggestions for other truths which could turn out to be local?