  The main motivation of this paper is the following open problem: Is the
hypercontractivity of the \emph{complex} polynomial Bohnenblust--Hille
inequality an optimal result? We show that the solution to this problem has a
close connection with the searching of the optimal constants for the
\emph{real} polynomial Bohnenblust--Hille inequality. So we are lead to a
detailed study of the hypercontractivity constants for real scalars. In fact we
study two notions of constants of hypercontractivity: absolute
($H_{a,\mathbb{R}}$) and asymptotic ($H_{\infty,\mathbb{R}}$). Among other
results, our estimates combined with recent results from \cite{CMPS} show that
\[ 1.5098<H_{\infty,\mathbb{R}}<2.829 \quad \text{and} \quad
1.6561<H_{a,\mathbb{R}}<3.296. \]
