In Peskin and Schroeder chapter 2 they discuss the quantization of the real Klein-Gordon field. First, they do this in the Schrodinger picture and they assert: $\phi(x)$ and $\pi(y)$ obey the canonical commutation relations, and they do not depend on time.
They write: $$\phi(x) \propto \int (a_{\ \vec{p}}\ e^{i \vec{p} \cdot \vec{x}} + a^\dagger_{\ \vec{p}}\ e^{-i \vec{p} \cdot \vec{x}}), $$ $$\pi(x) \propto \int (a_{\ \vec{p}}\ e^{i \vec{p} \cdot \vec{x}} - a^\dagger_{\ \vec{p}}\ e^{-i \vec{p} \cdot \vec{x}}) .$$
In the next section, they switch to the Heisenberg picture and make the operators time dependent: $$\phi(x) \to \phi(x,t) = e^{iHt} \phi(x) e^{-iHt}, $$ $$\pi(x) \to \pi(x,t) = e^{iHt} \pi(x) e^{-iHt}.$$ With the result $$\phi(x) \propto \int (a_{p}\ e^{-i p \cdot x} + a^\dagger_{p}\ e^{-i p \cdot x}), $$ $$\pi(x) = \frac{\partial}{\partial t} \phi(x,t).$$
This is all well and good, but how are we supposed to reconcile $\phi(x)$ having no time dependence in the Schrodinger picture? For example, by definition in the Lagrangian density we started with, $$L = (1/2) \dot{\phi}^2 - (1/2) (\nabla \phi)^2 - (1/2) m^2 \phi^2 $$ the field $\phi$ has time dependence, and $\pi(x) \equiv \frac{\partial L}{\partial \dot{\phi}} = \dot{\phi}.$
When we go on to quantize the complex scalar field, if we use the time dependent operators we can correctly guess the form $\pi$ should take by using $\pi = \dot{\phi}.$ But shouldn't we be able to stick to the Schrodinger picture? If we do this, it is easy to make the "incorrect" guess for the form of $\pi.$
Furthermore, we demand the canonical quantization relations: $$[\phi(x), \pi(y)] = i \delta^{3}(x-y) = [\phi^*(x), \pi^*(y)],$$ but shouldn't $$[\phi^*(x), \pi^*(y)]=[\phi(x), \pi(y)]^* = -i \delta^{3}(x-y)?$$
