I am reading Peskin & Schroeder and in chapter 2 (p.21) he quantizes the real K-G field such as: $$\phi=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_p}}\left(a_pe^{ip·x}+ a^{\dagger}_pe^{-ip·x}\right)$$ in analogy with the harmonic oscillator.
I understand that this allows $\phi=\phi^{\dagger}$ as it should.
The problem arises in exercise 2.2.b) when you are asked to follow the same procedure but for a Complex K-G field and "show that the theory contains two sets of particles of mass m". In this case I don't think the first expression holds true since $\phi=\phi^{\dagger}$ is not always the case. I've seen that the solution is to express the field in terms of two different creation and annihilation operators $a$ and $b^{\dagger}$ in the following way:
$$\phi=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_p}}\left(a_pe^{-ip·x}+ b^{\dagger}_pe^{ip·x}\right)$$
But I don't undersand why. What is the condition on the complex field that forces you to introduce $b^{\dagger}$ ? Is it just to ensure that $\phi$ is different than $\phi^{\dagger}$ ? Later on this b's turn out to be the ladder operators for antifermions, that come as a negative-frequency solution of the Dirac equation, but I don't see the relation between this and my K-G complex field. Can anyone shed some light in this issue?