You can split up the real and imaginary part of the state into their own seperate states. Namely by defining $x_r=\mathrm{Re}(x)$, $x_i=\mathrm{Im}(x)$, $A_r=\mathrm{Re}(A)$ and $A_i=\mathrm{Im}(A)$ then the differential equation can also be written as

$$ \dot{x}_r+i\,\dot{x}_i=(A_r+i\,A_i)(x_r+i\,x_i)+B\,u $$

which when split into their real and imaginary part gives

$$ \begin{bmatrix} \dot{x}_r \\ \dot{x}_i \end{bmatrix}= \begin{bmatrix} A_r & -A_i \\ A_i & A_r \end{bmatrix} \begin{bmatrix} x_r \\ x_i \end{bmatrix}+ \begin{bmatrix} \mathrm{Re}(B\,u) \\ \mathrm{Im}(B\,u) \end{bmatrix}. $$

The output can be expressed using

$$ y=C\begin{bmatrix} I & i\,I \end{bmatrix} \begin{bmatrix} x_r \\ x_i \end{bmatrix} + D\,u. $$

You can split up the real and imaginary part of the state into their own seperate states. Namely by defining $x_r=\mathrm{Re}(x)$, $x_i=\mathrm{Im}(x)$, $A_r=\mathrm{Re}(A)$, $A_i=\mathrm{Im}(A)$, $u_r=\mathrm{Re}(u)$, $u_i=\mathrm{Im}(u)$, $B_r=\mathrm{Re}(B)$ and $B_i=\mathrm{Im}(B)$ then the differential equation can also be written as

$$ \dot{x}_r+i\,\dot{x}_i=(A_r+i\,A_i)(x_r+i\,x_i)+(B_r+i\,B_i)(u_r+i\,u_i) $$

which when split into their real and imaginary part gives

$$ \begin{bmatrix} \dot{x}_r \\ \dot{x}_i \end{bmatrix}= \begin{bmatrix} A_r & -A_i \\ A_i & A_r \end{bmatrix} \begin{bmatrix} x_r \\ x_i \end{bmatrix}+ \begin{bmatrix} B_r & -B_i \\ B_i & B_r \end{bmatrix} \begin{bmatrix} u_r \\ u_i \end{bmatrix}. $$

The output can be expressed using

$$ y=C\begin{bmatrix} I & i\,I \end{bmatrix} \begin{bmatrix} x_r \\ x_i \end{bmatrix} + D\begin{bmatrix} I & i\,I \end{bmatrix} \begin{bmatrix} u_r \\ u_i \end{bmatrix}. $$