  We show that there exist infinite-dimensional extremely non-complex Banach
spaces, i.e. spaces $X$ such that the norm equality $\|Id + T^2\|=1 + \|T^2\|$
holds for every bounded linear operator $T:X\longrightarrow X$. This answers in
the positive Question 4.11 of [Kadets, Martin, Meri, Norm equalities for
operators, \emph{Indiana U. Math. J.} \textbf{56} (2007), 2385--2411]. More
concretely, we show that this is the case of some $C(K)$ spaces with few
operators constructed in [Koszmider, Banach spaces of continuous functions with
few operators, \emph{Math. Ann.} \textbf{330} (2004), 151--183] and [Plebanek,
A construction of a Banach space $C(K)$ with few operators, \emph{Topology
Appl.} \textbf{143} (2004), 217--239]. We also construct compact spaces $K_1$
and $K_2$ such that $C(K_1)$ and $C(K_2)$ are extremely non-complex, $C(K_1)$
contains a complemented copy of $C(2^\omega)$ and $C(K_2)$ contains a
(1-complemented) isometric copy of $\ell_\infty$.
