yeah, my initial thought on reading it in Monks & Palmer was that both theories seem pretty implausible. Reading the actual paper made it seem a bit more reasonable, though, and it also seemed equally implausible that argonauts would develop an egg case that's so similar in shape to a shell found in other cephs yet not in many other critters.
Monks & Palmer introduce the topic with this observation:
The most curious thing about the argonaut is its egg case, particularly the fact that it closely resembles an ammonite shell , such as the Triassic ammonite Trachyceras. This is unlikely to be the result of convergent evolution, that is, a similar solution to a common problem, because the ammonite's shell and the argonaut's do completely different things. Could it be mere coincidence?
Come to think of it, since argonauts are free-swimmers that sometimes float on the surface, how to they get positive buoyancy? They have no phragmacone; most other cephs are either negative buoyant or ammoniacal, but I thought only squids and maybe
vampyroteuthis had developed the ammonia buoyancy mechanism... are argonauts an example of that in octopods? Actually, Nixon & Young point out that "The female is neutrally buoyant, aided by a bubble of gas trapped in the upper part of the shell (Young 1960a, Voss and Williamson 1971, Roper 1972)" so arguments that the "egg case" is functionally distinct from proper phragmacones may be exaggerated... (although they don't mention that the animal can replenish the gas by some mechanism, or where the gas comes from in the first place... but they sink if you take it away).
Anyway, the Lewy paper addresses some of the questions a bit more. Part of his thesis is that ammonite macrononchs (presumed to be female) often have strange and restrictive-seeming body chambers at sexual maturity, so he hypothesizes that female ammonites, at the end of their lives, would seal themselves off to raise their young who would consume the mother in some sort of oedipal cannibalism feast when they hatch. That would explain another complaint about the theory, which is that the argonaut egg case is secreted from the arms, and is a different material than ammonite or nautiloid shells, which were presumably secreted from the mantle. It would also explain why heteromorph ammonites often have a final growth stage that seems to actively interfere with the body's orientation to do anything sensible.
Material-wise, ammonite shells are aragonite, and their aptychi are calcite. The Monks and Palmer book suggests that argonaut egg cases are calcite, but Hallucigenia found a ref that it's chitin:
argonauts use chitin! it's not CaCO3 at all!
and also notes that this info about argonauts:
wikipedia says males semelparous, females iteroparous
would seem inconsistent with Lewy's idea of argonauts sealing themselves off fatally as part of the brooding process..
Nixon & Young confirm that females brood for most of their lives, and say the shell is calcareous and structurally the greater part of it is finely prismatic, consisting of a thicker upper and a thinner under layer, between which is a thin layer of very fine, irregularly grained calcite (Boggild 1930, Noda et al. 1986).
Unfortunately, I had to leave when we were still discussing this, so I didn't really get her final thoughts on it...
One seeming flaw is that Lewy doesn't really explain why an argonaut egg case would look like the main ammonite shell, since if it evolved from some separate thing that was supposed to seal off the shell opening, why would it look anything like the shell?
I think it also raises another question, though: is there some aspect of cephalopod developmental biology control that favors mechanisms that produce spiral things? Such that when the argonaut needed to develop an egg case, it pulled some "spiral pattern" out of its genetic toolbox, even though it's applied through arms instead of mantle, and with a different material.
I agree with Monks and Palmer that it seems awfully weird that it would happen by chance, since we don't see too many spiral animal forms. It's not just spiral, either; in addition to the general spiral shape, its streamlining and ornamentation are rather typical of ammonites and not too common elsewhere. I can't think of a gastropod shell that is as similar, and I can't think of many other spirals in the animal kingdom at all, except maybe some brain structures, the cochlea, and rams' horns. (Although fans of the spirals and the golden ratio like to list nautilus as being kinda close to a "golden spiral," although measurably it doesn't seem locked to the phi, and fossil forms certainly aren't even though planispiral shelled cephs are closer to non-golden logarithmic spirals than anything else.)
One last comment: aren't argonauts extremely rare fossils, since their egg cases aren't much easier to preserve than soft-bodied animals? We seem to infer a lot of coleoid history from very few fossils...
edit: I removed the chitin note in wikipedia, since I found 3 sources that said it was wrong