Charge conjugation of fields This page on Wikipedia says, "In the language of quantum field theory, charge conjugation transforms as -

*

*$\psi \Rightarrow -i\big(\bar{\psi} \gamma ^0 \gamma ^2 \big)^T $

*$\bar{\psi} \Rightarrow -i\big(\gamma ^0 \gamma ^2 \psi \big)^T $ 
Could someone explain the reason behind choosing this specific mathematical transformation for charge conjugation (since the only thing it's supposed to do is invert the sign of all charges to convert a particle into its anti-particle)? Any relevant sources are also welcome.
 A: You want to implement the symmetries as transformations on the fields (and also corresponding operators on the Hilbert-space in the QFT), i.e. transformations of the dynamical variables in your Lagrangian $\mathcal{L}(\psi, \bar{\psi},...)$. The reason is that these symmetries of the Lagrangian have important consequences: Continuous symmetries for example give classically conserved current through Noethers theorem and Ward identities in the quantum theory. Discrete symmetries, like $C,P,T$ are also important.
Now let us apply the charge conjugation, which you defined above, to the QED Lagrangian. One can see that
$\bar{\psi} \partial_{\mu} \gamma^{\mu} \psi$ and $\bar{\psi} \psi$ are invariant, but $\bar{\psi} \gamma^{\mu} \psi \rightarrow - \bar{\psi} \gamma^{\mu} \psi$, so the QED interaction term transforms as
$$
e A_{\mu} \bar{\psi} \gamma^{\mu} \psi \rightarrow - e A_{\mu} \bar{\psi} \gamma^{\mu} \psi,
$$
which is the same as taking $e \rightarrow -e$. This is one easy way of seeing why charge conjugation is an appropriate name. Now if we let $A_{\mu} \rightarrow - A_{\mu}$ under $C$ we get a symmetry of the QED Lagrangian and a consequence is for example Furry's theorem.
I also strongly recommend you to read p.192-195 in Schwartz's QFT book, where the explanation is more detailed and there is a more correct justification of the name charge conjugation.
A: The charge conjugation symmetry is the proper time reversal symmetry.
It becomes an exact symmetry of relativistic quantum systems if we combine with space-time time reversal and parity, due to unitarity.
About the techinical details you are raising I think that this transformation looks complicated just in you notation, afterwards you are hiding the indices.
I think that in four dimensions there is a suitable choice of indices ($SL(2,\mathbb{C})$ indices) where the charge conjugation reduces to:
$$\psi^{\alpha}\leftrightarrow \overline\psi^{\dot\alpha}\,\quad \alpha=1,2\;\quad \dot\alpha=3,4\,.$$
