Is the word "I" a dangling linguistic pointer with no actual referent? Otherwise, what exactly is this slippery referent? Could it be possible the phrase "I am" is meaningless due to a lack of an actual referent?
I would start answering this by pointing out that, even if it is syntactically meaningless, it would be dangerous to assume it is semantically meaningless. Wars have been fought over those two words, expressed in many many languages.
In the most basic parts of language, syntax catches up with semantics. When forming a language, the need for concepts proceeds the creation of the syntax (at least it appears to do so for the early phrases). Virtually all cultures (if not literally all cultures) support the concept of a Self (which can be exactly as slippery as you worry it is). This concept proves useful enough that it was given a single-syllable utterance (whether as a word "I" or verb conjugation).
Consider: entire schools of philosophy have arisen to combat the question of "what is the self?" because the "self" is such a useful semantic concept to have in virtually everything we do, and yet it does prove particularly difficult to pin down to a syntactic definition.
Well, if utter the word 'I', it's quite clear what it refers to: sequitur. If I utter 'I am', it's also quite clear what I say with my utterance: something that is true iff sequitur exists. On the other hand, if David Kaplan utters 'I am', he says something that is true exactly if David Kaplan exists. So, no problem here: 'I' is an indexical, whose referent in any context of utterance is simply the speaker of the context. Going intensional one might also say that the intension of 'I' in some context c is simply the constant function that assigns to each possible world (or index more generally) the speaker of c. (The referent of 'I' in c is then what the intension of 'I' in c assigns to the world of the context.)
What might be confusing is the fact that Kaplan's and sequitur's uses of 'I' have something in common, a sort of standing meaning, that does not vary with context. This can't be one of the intensions Kaplan and sequitur express with their uses of 'I', since in your context 'I' does express neither the one nor the other.
Rather, this standing meaning is a rule that tells us how to get from a given context to 'I''s intension in that context. This rule can be represented by a function from contexts to intensions, which can be described as follows: It assigns to any context c the (constant) intension which assigns to any world the speaker of c. Kaplan calls this function the character of 'I'.
So now we have to layers of meaning for 'I': 'I''s contextually invariant character (operating on contexts) and 'I''s intension in a context (operating on worlds). Your confusion arises from trying to treat 'I''s character as it were 'I''s intension in a context. 'I''s intension in a context returns a definite referent for any world as argument (the context's speaker). The character, however, returns intensions which have to be evaluated in worlds.
Ernst Mach, a physicist and philosopher, was a proponent of the view that
there is no such thing as the self, the "I" and therefore that "I" has, strictly speaking, no referent.
(His theory also gives a explanation in terms of proto-evolutionary psychology of why we ordinarily think there is such a thing as the self.)
He argues this at length in the first chapter of his book The Analysis of Sensations (Die Analyse der Empfindungen, 1886/1905) which expounds his "Elemententheorie" (theory of elements) based on psychological and 'sociobiological' findings of his time.
Here is his memorable conclusion (IMO in a terrible English translation):
The primary fact is not the ego, but the elements (sensations). What was said on p. 21 as to the term " sensation " must be borne in mind. The elements constitute the I. That I have the sensation green signifies that the element green occurs in a given complex of other elements (sensations, memories). When I cease to have the sensation green, when I die, then the elements no longer occur in the ordinary, familiar association. That is all. Only an ideal mental-economical unity, not a real unity, has ceased to exist. The ego is not a definite, unalterable, sharply bounded unity. None of these attributes are important; for all vary even within the sphere of individual life; in fact their alteration is even sought after by the individual. Continuity alone is important. This view accords admirably with the position which Weismann has reached by biological investigations. […] But continuity is only a means of preparing and conserving what is contained in the ego. This content, and not the ego, is the principal thing. This content, however, is not confined to the individual. With the exception of some insignificant and valueless personal memories, it remains presented in others even after the death of the individual. The elements that make up the consciousness of a given individual are firmly connected with one another, but with those of another individual they are only feebly connected, and the connexion is only casually apparent. Contents of consciousness, however, that are of universal significance, break through these limits of the individual, and, attached of course to individuals again, can enjoy a continued existence of an impersonal, superpersonal kind, independently of the personality by means of which they were developed. To contribute to this is the greatest happiness of the artist, the scientist, the inventor, the social reformer.
The ego must be given up ["Das Ich ist unrettbar", lit. "The I cannot be rescued"]. It is partly the perception of this fact, partly the fear of it, that has given rise to the many extravagances of pessimism and optimism, and to numerous religious, ascetic, and philosophical absurdities. In the long run we shall not be able to close our eyes to this simple truth, which is the immediate outcome of psychological analysis. We shall then no longer place so high a value upon the ego, which even during the individual life greatly changes, and which, in sleep or during absorption in some idea, just in our very happiest moments, may be partially or wholly absent.
(The Analysis of Sensations, Ch. 1 slightly amended, emphasis added)
You can read/download the whole book in English translation for free (public domain):
- The Analysis of Sensations and the Relation of the Physical to the Psychical (translated by Williams and Waterlow, Open Court, 1914).
More on Ernst Mach:
Short Answer... "I" has a referent, albeit not a concrete one.
"I" can refer to a variety of things. For instance, is "I" referring to the same thing in the following statements?
- I am upset.
- I will kick the ball.
In the first, "I" seems to point to a feeling agent. In the second, it seems to point to a willing agent, or perhaps even the physical one.
On the other hand, "I" may be a logical fiction, as the hypothetical identity or substance behind the feeling, willing and physical.
However, because it's a logical fiction doesn't mean it has no referent; it's just that its referent is an abstraction.
A lot has been written on this, although much of it is in the Eastern tradition (like Buddhism). On the more (Western) philosophical front, Bundle Theory may be a good start, and Derek Parfitt has done long treatments of this subject.
As an aside, the case of "I" being a logical fiction is actually a specific case of a more general negation of substance ontology. That is, Bundle Theorists would point out that there is no identity beyond the perceptions we have of things, we simply give them identity as logical fictions, and the "I" is a particular instance of this, wherein we give an identity to our sensations, thoughts, possessions, etc...