  The concept of $\mathbb C$-supercyclic operators was introduced by Hilden and
Wallen in \cite{hilden} and, since then, a multitude of variants have been
studied. In \cite{herzog1992linear}, Herzog proved that every real or complex,
separable, infinite dimensional Banach space supports a $\mathbb C$-supercyclic
operator. In the present paper we investigate different variants of
supercyclicity, precisely $\mathbb R^+$-, $\mathbb R$- and $\mathbb
C$-supercyclicity in the context of composition operators. We characterize
$\mathbb R$-supercyclic composition operators on $L^p$, $1 \leq p < \infty$.
Then, we turn our attention to dissipative composition operators, and we show
that $\mathbb R$- and $\mathbb C$-supercyclicity are equivalent notions in this
setting and they have a ``shift-like'' characterization.
