Dear Francesco,
Analytic functions and fields, in particular holomorphic functions, have the property that if you know all the (partial) derivatives at any point, they determine the value at any other point. Take for example a real analytic function like a polynomial P(x), where x is a real variable. The derivatives at x=0 allow you to determine the coefficients of the polynomial, and then you can determine P(x) for any other point x. And this works as well with the exponential, sin, cos, etc.
Now complex holomorphic functions also are analytic, but they are more than this. So why I picked them and not just analytic functions, which would give me more chances to prove this? It is because equations in mathematical physics can have in general non-analytic solutions, but in the case of the Cauchy-Riemann the solutions can't be non-analytic.
Holomorphic fundamentalness means that everything about the field is encoded at any point of spacetime you choose, but the same data is encoded, differently, at any other point. Here is where the relativity of such fundamentalness is, because there is no special point of spacetime. Now why I say this relativity of fundamentalness is not merely epistemological, but also ontological? Because no matter what is the ontology of your theory, that ontology at a single point contains everything. But since you are free to choose the point, the ontology at any other point also contains everything, encoded of course differently, because the partial derivatives of your fundamental field at the other point are different. So, if there is an ontology, and this is governed by universal laws, that is, they are the same everywhere (which is the basic assumption in physics), and if the fields describing the ontology are analytic, they will have this property of ontological holomorphic fundamentalness. In fact, I think the epistemological aspect will not work in practice, because you can't know all the derivatives at a given point.
Note that by "ontology" I don't mean what one usually means, some stuff that is like what we think matter is from our daily experience. I see it more like a mathematically consistent stuff that is real but who's state is not defined prior to observations, and it becomes more and more defined as we add more (quantum) observations. But I also see it in the ways I described in this essay. I couldn't expand much this view in this essay, but I introduced the ideas gradually in the previous essays. This feature of reality of becoming gradually defined (but not gradually real) is in these essays Flowing with a Frozen River and The Tao of It and Bit. The latter introduces a bit to how I see the mathematical universe, which I expanded more in And the math will set you free. And in The Tablet of the Metalaw I continued a bit more with some ideas related to the hard problem of consciousness and some relative ontology which I will discuss later. What I wrote here is how I view the things for cca 25 years, but I introduced them gradually in my FQXi essays, which can be seen both as a series and as stand-alone.
About Leibniz's monads and Nagarjuna's relativity, I guess we can make some connections, but I don't want to claim they are all the same. In fact, even the "Indra's net" idea I see it more like a metaphor, because I can't say that it meant exactly the same (and I only found out about it many years later I had the idea of holomorphic fundamentalness, when I told it to a friend and he mentioned Indra's net). Nagarjuna's relativity of existence vs nonexistence, where he states that all things exist/don't exist/both exist and don't exist/neither exist nor don't exist, can be compared to some things I said or where said in physics. While a common understanding is that he was an anti-realist, I think this is an oversimplification. So here are some possible connections. It can be applied to wave-particle duality in quantum mechanics. But I would rather apply it to the existence of a definite state of reality. I would apply it to the idea I explored in previous essays, that the universe is mathematical (in a sense which I tried to make clear how is different from Tegmark's). In The Tablet of the Metalaw, section "Why is there something rather than nothing", I propose that the only things that exist are those which can't not exist (here it can be paralleled with Leibniz's Contingency argument, but I think there are some essential differences), and that only mathematics and logic have this feature. But I can't say that subjective experience is reducible to mathematics, so I would say rather that there is some thing that is mathematical but also semantic, in the sense that it has embedded the 'sentience' at the origin of our subjective experience, for which the mathematical structure has the meaning of a universe. Also relativity of fundamentalness, which is both epistemological, and ontologic in the sense I mentioned, also can be connected with Nagarjuna's in some sense: if only what's fundamental exists, and what is derived doesn't, then his relativity is entailed by mine because I claim that the distinction between what's fundamental and what's derived is relative. But such ideas are open to interpretations, so I only want to point out some connections, not to claim that what I said confirmed what he said or the other way around.
Best regards,
Cristi