A first (non-archimedean) step in this direction was given by Bhatt in his paper on the derived approach to p-adic Hodge theory (originally due to Beilinson), where he proposed a Hodge theory for (say semi-stable) varieties over $\mathbb{Z}$, using Illusie’s derived de Rham cohomology? to define a convenient period ring?. From the point of view of global analytic geometry, this gives a Hodge theory for strict analytic spaces over $\mathbb{Z}$, equipped with its trivial norm. One may try to adapt these ideas to extend them to a Hodge theory for strict analytic spaces over $\mathbb{Z}$, equipped with its archimedean norm, but the use of a derived completion in the process prevents one from a straightforward generalization. Remark that the treatment of quasi-projective varieties should certainly involve a global mixed Hodge theory for pre-proper logarithmic analytic varieties.

The advantage of including the archimedean norm in global Hodge theory is that this seems to give a better setting for the study of completed arithmetic L-functions? and automorphic L-functions, whose functional equation may not be proved without using the archimedean information given by a convenient Gamma-factor. This extension is very easy to make if one works with rational coefficients, like in Beilinson’s original paper on p-adic Hodge theory (i.e., with the de Rham period ring). It remains a challenging problem to combine the non-archimedean and archimedean information on Hodge theory without forgetting to take care of torsion phenomena in cohomology, that are very important for the study of special values of L-functions. This may be doable if one follows the line of ideas used in Arakelov geometry, but formulated in a global analytic way. A very natural setting for this seems to be global strict analytic geometry over $\mathbb{Z}$ equipped with its archimedean norm.