Sorry for the lengthy question, pretty much the whole text is the standard derivation of the solution of the KG equation which I included to illustrate my doubts, and some questions are at the end. The Klein-Gordon equation is
$$ (\partial^2+m^2)\phi(x)=0\tag{1}$$ where $\partial^2=\partial_\mu\partial^\mu$. Taking a Fourier transform of $\phi$
$$\phi(x)=\int \frac{d^4 k}{(2\pi)^4}\phi(k)e^{ikx} \tag{2}$$
and inserting it into the equation we have
$$ \int \frac{d^4 k}{(2\pi)^4}(k^2-m^2)\phi(k)e^{ikx}=0\tag{3}$$
which implies
$$(k^2-m^2)\phi(k)=0 \implies \phi(k)=2\pi f(k)\delta(k^2-m^2) \tag{4}$$
for some function $f$. Define $\omega=\sqrt{\mathbf{k}^2+m^2}$ where $k=(k_0, \mathbf{k}$), then $$\delta(k^2-m^2)=\frac{1}{2\omega}\left[\delta(k_0-\omega)+\delta(k_0+\omega)\right]\tag{5}$$
And our solution becomes
$$\phi(x)=\int \frac{d^4 k}{(2\pi)^3}\frac{1}{2\omega}\left[\delta(k_0-\omega)+\delta(k_0+\omega)\right]f(k)e^{ikx}\tag{6}$$
performing the integration on $k_0$
$$\phi(x)=\int \frac{d^3 k}{(2\pi)^3}\frac{1}{2\omega}(f(\omega,\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}+i\omega t}+f(-\omega,\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}-i\omega t})\tag{7}$$
In the second term, we can change variable $\mathbf{k}\rightarrow-\mathbf{k}$ and get
$$\phi(x)=\int \frac{d^3 k}{(2\pi)^3}\frac{1}{2\omega}(f(\omega,\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}+i\omega t}+f(-\omega,-\mathbf{k})e^{-i\mathbf{k}\cdot\mathbf{x}-i\omega t})\tag{8}$$
Now: all the sources I can find go on and impose that the field must be real, and write something like
$$\phi(x)=\int \frac{d^3 k}{(2\pi)^3}\frac{1}{2\omega}(a(\omega,\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}+i\omega t}+a^\dagger(\omega,\mathbf{k})e^{-i\mathbf{k}\cdot\mathbf{x}-i\omega t})\tag{9}$$
and this is presented as the "general solution to the Klein-Gordon equation"
Questions:
Why should we impose that the field is real, i.e. $f(-\omega, -\mathbf{k})=f(\omega, \mathbf{k})^\dagger$? I see no mathematical reason to do so, and I don't see why a complex field would pose physical problems. I know about the harmonic oscillator interpretation, but I'd say that this interpretation is a consequence of the fact that the field is real and not vice versa. Why do we eliminate complex field solutions?
Are the $2\pi$ factors important? In equation $(4)$ I introduced a factor $2\pi$ just so that the final solution would coincide with the one given in standard sources, but again, I don't see another reason to add it (actually the whole "the solution must be a function times delta" is a bit sketchy, how to see this?)
Some give the solution with $\sqrt{\omega}$ instead of $\omega$ in the denominator (Peskin & Schroeder for instance) in analogy with the harmonic oscillator. To try and get this I thought of defining $a(\omega,k)=\sqrt{\omega}f(\omega,k)$ instead of $a(\omega,k)=f(\omega,k)$. Does this make sense? Do the two solutions have any physical difference?
