  We define the universal thickening of the field of real numbers. This
construction is performed in three steps which parallel the universal
perfection, the Witt construction and a completion process. We show that the
transposition of the perfection process at the real archimedean place is
identical to the "dequantization" process and yields Viro's tropical real
hyperfield. Then we prove that the archimedean Witt construction in the context
of hyperfields allows one to recover a field from a hyperfield, and we obtain
the universal pro-infinitesimal thickening of the field of real numbers. We
provide the real analogues of several algebras used in the construction of the
rings of p-adic periods. We supply the canonical decomposition of elements in
terms of Teichmuller lifts, we make the link with the Mikusinski field of
operational calculus and compute the Gelfand spectrum of the archimedean
counterparts of the rings of p-adic periods. In the second part of the paper we
discuss the complex case and its relation with the theory of oscillatory
integrals in quantum physics.
