  We introduce a new invariant for a $2$-knot in $S^4$, called the
shadow-complexity, based on the theory of Turaev shadows, and we give a
characterization of $2$-knots with shadow-complexity at most $1$. Specifically,
we show that the unknot is the only $2$-knot with shadow-complexity $0$ and
that there exist infinitely many $2$-knots with shadow-complexity $1$.
