  A finite set $X$ in a complex sphere is called a complex spherical $2$-code
if the number of inner products between two distinct vectors in $X$ is equal to
$2$. In this paper, we characterize the tight complex spherical $2$-codes by
doubly regular tournaments, or skew Hadamard matrices. We also give certain
maximal 2-codes relating to skew-symmetric $D$-optimal designs. To prove them,
we show the smallest embedding dimension of a tournament into a complex sphere
by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel
matrix.
