Non-algebraic Hecke characters Algebraic Hecke characters are ubiquitous in modern number theory. They are in 1-1 correspondence with one dimensional complex Galois representations, and in some precise sense they are the building blocks of everything with complex multiplication, up to the motivic level.
But it is an observation of Weil (Sur le theorie du corps de classes, 1951) that not all Hecke characters (that is, automorphic forms for $\mathrm{GL}_1(\mathbb{A})$) correspond to Galois representations. The Weil group was later defined as an extension of the absolute Galois group that contains the non-algebraic Hecke characters too.
This phenomenon gets more complicated and drastic in higher dimensions, but the basic idea still is that some automorphic forms, denoted algebraic, have obvious number theoretic meaning (they capture information coming from motives), and some of them don't.
Another basic example of this kind of behavior (this time in $\mathrm{GL}_2(\mathbb{A})$) are Maass forms: those with eigenvalue $1/4$ (and only those) are expected to correspond to Galois representations.

What I am trying to understand is, is there any result or
  philosophy as to what information non-algebraic Hecke characters
  contain?

Every time they are mentioned one gets the impression that the fact that they don't contain motivic data makes them uninteresting. Perhaps there is a good reason for this, but on the outset is seems the same as saying that $e$ is not interesting because it isn't algebraic.
The same question could be asked about trascendental automorphic forms in general, but this relatively simple and concrete case seems vague and broad enough.
 A: Here is a highly conjectural guess, which I hope some experts can confirm or deny. Kontsevich and Zagier in their paper Periods define an algebra of periods, defined by integrals of algebraic equations over (semi-)algebraic domains in $\bf R^n$. In the final section Kontsevich claims that the ring of extended periods (where $\pi$ is inverted) is a (pro-algebraic) torsor for the motivic Galois group, i.e., the Tannakian fundamental group of the category of pure motives, say, over $\bf Q$. This ring of periods can be further extended to include 'exponential periods,' which have been explored by Bloch and Esnault, among others.
The point here is that Langlands conjectures a homomorphism from this motivic Galois group into the conjectural automorphic Langlands group, i.e., the Tannakian fundamental group of the category of isobaric automorphic forms of $GL_n(\bf Q)$ for all $n$, which is not known to exist in general.
A possibility is the following: the automorphic Langlands group should, in this context, be a torsor for a larger ring of periods, in some way related to this extended ring of periods. It ought to be then that the periods of non-algebraic Hecke characters and Maass forms of eigenvalue greater than $1/4$ belong here.
In some sense, this is just a very fancy way of saying non-algebraic Hecke characters are interesting precisely because they are not algebraic, the way that $e$ is interesting because it is not algebraic.
