Decoupled physics of the complex scalar field The canonical commutation relations for a complex scalar field are of the form
$$[\phi(t,\vec{x}),\pi(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})$$
$$[\phi^{*}(t,\vec{x}),\pi^{*}(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})$$
How can these commutation relations be obtained from the commutation relations for two free real scalar fields?
 A: I) The complex scalar field comes from two real/Hermitian scalar fields with equal-time CCRs
$$[\hat{\phi}^j(t,\vec{x}),\hat{\phi}^k(t,\vec{y})]~=~0, $$
$$[\hat{\phi}^j(t,\vec{x}),\hat{\pi}_k(t,\vec{y})]~=~i\hbar{\bf 1}~\delta^j_k~ \delta^{3}(\vec{x}-\vec{y}), $$
$$[\hat{\pi}_j(t,\vec{x}),\hat{\pi}_k(t,\vec{y})]~=~0, \qquad j,k~\in~\{1,2\}, \tag{A}$$
and the definitions 
$$ \hat{\phi}~=~\frac{1}{\sqrt{2}}(\hat{\phi}^1+i\hat{\phi}^2),\tag{B}$$
$$  \hat{\pi}~=~\frac{1}{\sqrt{2}}(\hat{\pi}_1\color{red}{-}i\hat{\pi}_2),\tag{C}$$ 
cf. e.g. this Phys.SE post. This leads to OP's mentioned CCRs.
II) If the $\color{red}{\text{minus sign}}$ in eq. (C) seems strange, consider the following classical argument. The Lagrangian density is
$$ {\cal L}~=~|\dot{\phi}|^2 - |\nabla \phi |^2 - {\cal V} 
~=~\frac{1}{2}(\dot{\phi}^1)^2+\frac{1}{2}(\dot{\phi}^2)^2-\frac{1}{2}(\nabla \phi^1)^2-\frac{1}{2}(\nabla \phi^2)^2  - {\cal V} .\tag{D} $$
Therefore the momenta read
$$ \pi_j~=~\frac{\partial {\cal L}}{\partial \dot{\phi}^j}~=~\dot{\phi}^j, \qquad j~\in~\{1,2\}, \tag{E}$$
$$ \pi~=~\frac{\partial {\cal L}}{\partial \dot{\phi}}
~=~\frac{1}{\sqrt{2}}\left(\frac{\partial {\cal L}}{\partial \dot{\phi}^1}\color{red}{-}i \frac{\partial {\cal L}}{\partial \dot{\phi}^2} \right)
~=~\dot{\phi}^{\ast}
~=~\frac{1}{\sqrt{2}}(\pi_1\color{red}{-}i\pi_2).\tag{F}$$
III) For reference, let us mention that the Hamiltonian Lagrangian density reads
$$ {\cal L}_H~=~\pi \dot{\phi}+\pi^{\ast} \dot{\phi}^{\ast} -{\cal H}
~=~\pi_1 \dot{\phi}^1+\pi_2 \dot{\phi}^2 -{\cal H}, \tag{G}$$
where the Hamiltonian density is
$$ {\cal H}~=~|\pi|^2 + |\nabla \phi |^2 + {\cal V}~=~\frac{1}{2}(\pi_1)^2+\frac{1}{2}(\pi_2)^2+\frac{1}{2}(\nabla \phi^1)^2+\frac{1}{2}(\nabla \phi^2)^2 + {\cal V} .\tag{H}$$
