A Banach space X is called extremely non-complex iff for every linear bounded operator T on X we have ||S+I||=1+||S|| where S is the square of T, i.e., the square of every linear operator satisfies the Daugavet equation.

We show that various kinds of infinite dimensional extremely non-complex spaces exist, they may even have complemented copies of c0 or l. We also investigate possible groups of surjective isometries on such spaces obtaining an example of a Banach space X where the only surjective isometries are I and -I, however the surjective isometries of the dual X* contain the group of isometries of an infinite dimensional Hilbert space.

Most examples are derived from C(K) spaces with few operators which seem to be more suitable for the isometric questions than Maurey-Gowers' spaces with few operators.