In many papers and books (as [1], [3]--[8] and others) the complex and almost complex structures defined on real spaces are examined. In this paper they are defined on complex Finsler spaces. The complex Finsler space $E'$ is formed in such a way, that its tangent space $T(E')$ is equal to $T(F_1)\oplus iT(F_2)$, where $F_1$ and $F_2$ are two $2n$-dimensional Finsler spaces. Using the nonlinear connections $N$ and $\bar N$ of $F_1$ and $F_2$ respectively, the adapted basis $B'$ of $T(E')$ is formed. In $T(E')$ different almost complex structures are given and the form of the corresponding Hermite metrics is determined.