Analytic motivic homotopy theory is an analytic generalization of motivic homotopy theory that gives back classical homotopy theory when one works with analytic varieties over $\mathbb{C}$ and motivic homotopy theory when one works with strict analytic varieties over a ring equipped with its trivial norm.

The proper setting for analytic motivic homotopy theory of quasi-projective varieties seems to be the setting of logarithmic motivic homotopy theory.

Joseph AyoubLa realisation de Betti et les six opérations (defines complex analytic motives and shows that they are equivalent to classical homotopy types)

Joseph AyoubMotives of rigid analytic varieties (defines rigid motives and uses them to study the functor of vanishing cycles on a characteristic $0$ “trait”).

Frédéric PaugamOverconvergent global analytic geometry (defines motives over an arbitrary base Banach ring)