  In this paper we study the notion of $C^{(n)}$-supercompactness introduced by
Bagaria in \cite{Bag} and prove the identity crises phenomenon for such class.
Specifically, we show that consistently the least supercompact is strictly
below the least $C^{(1)}$-supercompact but also that the least supercompact is
$C^{(1)}$-supercompact (and even $C^{(n)}$-supercompact). Furthermore, we prove
under suitable hypothesis that the ultimate identity crises is also possible.
These results solve several questions posed by Bagaria and Tsaprounis.
