The sentence "I am here now" does have some peculiar properties, at least insofar as it contains indexical expressions, which themselves are somewhat peculiar. Within the philosophy of language, the proper semantic treatment of indexicals is an area of live debate.
First off, what's an indexical expression? An indexical expression is, roughly, an expression whose reference can shift across contexts. Your sentence contains three such expressions: "I", "here", and "now". How does reference shift across contexts? Well, consider what the referent of "I" would be when you utter "I am here now" (hint: it's you). When I utter the same sentence, the reference of "I" shift to me. In usual circumstances (e.g., not in indirect discourse), "I" refers to the speaker.
We can see that this is an example of the reference of "I" shifting across contexts, where the role context plays is to specify the speaker of the sentence. This is why "I" is an indexical.
"Here" works similarly, except the thing referred to is (usually) the location of the speaker (fixed by the context). "Now" as well, with the thing referred to being the time of utterance (fixed by the context). David Lewis famously proposed an indexical analysis of "is actual", claiming that it functioned like "here", except that the location specified is the possible world inhabited by that individual (rather than a region within a possible world occupied by an individual).
Formally this all gets modeled using centered possible worlds. These were popularized (first discussed?) by David Lewis in his 1979 "Attitudes De Dicto and De Se". Centered worlds are possible worlds which come with privileged individuals and times. You can see why they'd be useful: context supplies some centered world as the world relative to which the utterance is to be evaluated (the "evaluation world") and this centered world privileges certain individuals and times in a way that makes them the default referents of indexical expressions (in that context). Location gets fixed because the privileged individual (assuming they aren't scattered individuals) occupies a location at the time of utterance in a way that makes them the default referents of indexicals (in that context).
So, back to your sentence "I am here now". When I utter that sentence, context selects a centered world where the privileged individual is me, the privileged location is where I am at the time of utterance, and the privileged time is when I am at the time of utterance (which is just a convoluted way of saying "it's the time of utterance"). When you utter the sentence the same thing happens, mutatis mutandis.
Contrast this with a sentence like "Dennis was in Massachusetts on October 14th, 2015". This sentence is true, but note that the reference of the terms "Dennis", "Massachusetts", and "October 14th, 2015" is (plausibly) the same no matter the context or who utters it (this, strictly speaking, depends on some plausible but controversial assumptions about proper names, e.g., that they're "rigid designators"). To evaluate it, you need only specify a possible world (by default, the actual world), and so you don't need the privileged individuals/times/locations supplied by centered worlds(/contexts).
The Answer to Your Question
So, to answer your question, what's peculiar with the sentence "I am here now" is that the the worlds relative to which you evaluate the utterance are centered worlds as opposed to the more familiar possible worlds of modal logic and formal semantics in the style of Heim and Kratzer (obviously excluding the sections on interpreting indexical expressions).
Indexicals: The Stanford Encyclopedia of Philosophy has (as usual) a great article on the subject of indexicals. The locus classicus here is David Kaplan's "Demonstratives". Kaplan's Theory of Indexicals is covered in the SEP article, but you can also read about it in the Internet Encyclopedia of Philosophy's entry on Demonstratives and Indexicals.
Possible Worlds in Formal Semantics: the standard text in formal semantics is Irene Heim and Angelika Kratzer's "Semantics in Generative Grammar" (copies of which can be found all over the internet). That's a good general reference on all of these topics.