I spent a long time wondering about this myself. Here are some metamathematical thoughts.
The idea of eigenanalysis is to find a way in which the action of a linear transform behaves like multiplication by a scalar, in the hope of simplifying the action of the transformation. For many transforms with real eigenvalues, this has neat interpretations. For a 2D rotation however, there isn't any neat interpretation as multiplication by real numbers. The easy extension of the idea uses complex eigenvalues -- whereby the rotation is exactly represented by complex multiplication by that eigenvalue, which itself has a geometric interpretation as the same rotation.
The fundamental theorem of algebra is equivalent to the existence of a complex eigenvalue of any mapping of complex n-space to itself. That is very neat. But is it the answer to all questions in linear algebra? (No, it's not.)
One question that might be asked here is: to what degree is an eigenanalysis of a quaternion useful? Another might be, what does an eigenanalysis even mean?
You're asking for eigenvalues of a transform equivalent to (left- or right-) multiplication by a quaternion. What exactly do you hope for? Well, real eigenvalues are usually too much to expect, given what we know of general real transformations. So ... could it be that a quaternion behaves like complex multiplication with some other quaternion? Well, it could. Besides quaternions that are simple scalings (which have a single real eigenvalue), some quaternions represent simple rotations of 3-space, and thus have complex and real eigenvalues. But not all quaternions behave so: typically they model a richer set of changes of orientation (a rotation and a twist).
But the whole point of the construction of the quaternions is to produce a four-dimensional entity that behaves somehow as a scalar. What is a "scalar" though? It's a matter of semantics: if the term means something that is only a "size", it rules out complex scalars. In recent decades the term has been extended to include the complex numbers, and specifically meant to exclude anything else -- specifically to accommodate things like eigenanalysis. But if the term means an element of an algebra that behaves somehow like numbers, say, element of a division ring, and you're not picky about commutativity, quaternions can be viewed as a sort of algebra of scalars.
Like the situation with eigenvalues of a complex number, the most satisfying interpretation of an "eigenvalue" for a typical quaternion may be: the quaternion itself.