As you may have noticed, grammatical alignments have really held my attention for a while now. In part because they really lend themselves to easy creativity, another reason is that there's a fair share of stuff that looks almost realistic yet might not be attested anywhere.
One thing I'd like to draw attention to - a thing I bet already has been considered by linguists somewhere - is the possibility of drawing analogies between adnominal possession and alignment and also between predicative possession and alignment.
Comparing nom-acc vs. erg-abs with head marking and dependant marking adnominal possession gets a bit unclear. If we assume the possessum is the head, the following comparisons obtain:
nominative: independent noun, possessed noun
accusative: marked possessor
ergative: marked possessum
absolutive: independent noun, possessor
However, we could argue for the other way around as well.
I have presented an inverse alignment for possession as well, but let's consider some other options:
Tripartite: each of the possible noun phrases - non-possessed, possessor, and possessum all have unique markings. To some extent, double-marked possession is exactly this.
We go on to consider some kind of split-possession system: maybe nouns in the "basic" cases require possessors in the genitive, whereas in adpositional phrases, the possessor OR the independent noun takes the case the adposition calls for, whereas a possessum takes some specific marking. Alternatively, nouns in the "basic" cases can also be marked as possessums instead of their usual case marking (c.f. how Finnish conflates possessed nominatives and accusatives), whereas oblique cases must be marked. The other option with split possession is to have it lexically determined: some nouns trigger head-marking, some trigger dependent-marking.
Once that is done, we get to the point of omitted arguments: what if we want to mark that a thing is possessed, but not explicitly state the possessor? What if we want to state the possessor, but not the possessum?
These too could permit for patterns analogous to anti-ergative, "normal accusative" and so forth. Some languages - including to some extent English - have a special case for this, viz. mine, yours, hers, ours, etc. Some languages have a special form for the opposite situation, i.e. basically [someone's] hand, for nouns that are cannot stand without possessors.
All of these could basically be taken over directly to the predicative possession situation: different case uses, different constructions (possessum is by possessor, possessor is with possessum, the possessum possibly being more or less syntactically subject- or object-like in either of the constructions, etc) ...
Let's, however, not stop here, let's ... go where no man has gone before:
What could a possessive analogue of ditransitives be?
How likely is it for a language to evolve some kind of regular, grammaticalized ditransitive type of possession? What use would it have? How could it, morphologically and syntactically, be formed?