2.0201 Each statement about complexes can be analyzed into a statement about their components and into those propositions that completely describe the complexes.
Black says (p. 61) cf. 3.24, 3.3442, and 5.5423. He goes on to note that in the corresponding part of Notes on Logic (ref. 99 (6) c) Wittgenstein adds “i.e. that proposition which is equivalent to saying the complex exists.” If this is what he means here, then the whole thing just means that each statement that a complex exists can be analyzed into a statement that this and this and this ingredient are combined thus into a complex.
My initial reaction: A bit mystifying again. What complexes? If I make a statement about three complexes ("Complexes A, B, and C are nice," say) then this can be analyzed into a statement about the components of A, B, and C, and into the set of propositions that describes A, B, and C completely? Wouldn't that require a lot more information? If A is a tiger can an analysis of my statement that it is nice give us a complete description of the tiger? Surely this is not what Wittgenstein means, but what he does mean is not apparent yet. Could a complex be whatever is taken for a complex, so that a statement about a complex would be about a complex-understood-as-a-complex, which could then be analyzed into a statement about the-things-understood-to-be-parts-of-the-complex plus some propositions about those-parts-considered-as-parts-of-a-complex? There might be limits to what could be thought of as a complex (perhaps) but otherwise complexity might be in the eye of the beholder. Or rather, in the analysis of the statement. Simplicity and complexity would then not be absolute matters, matters of metaphysics, but dependent on, or features of, logical analysis. And what that is is not (yet) clear.
Ostrow (p. 27) says that this remark must be compared with 3.24. The “central purpose” of 2.0201, he says (p. 28) is to make evident the fundamental distinction between complex and object. Complexes cannot be treated as entities or objects. By 2.0211, a proposition about a nonexistent object is nonsense, but, by 3.24, a proposition about a nonexistent complex is false, not nonsensical.