Suppose we have Weyl spinor $\psi_{a}$, which transforms under irreducible representation $\left( \frac{1}{2}, 0\right)$ of the Lorentz group,

$$

\psi_{a} \to (T(g))_{a}^{\ b}\psi_{b},

$$and complex conjugated spinor $\kappa^{\dot{b}}$, which transforms under $\left(0, \frac{1}{2}\right)$ representation,

$$

\kappa^{\dot{b}} \to (T(g))^{\dot{b}}_{\ \dot{a}}\kappa^{\dot{a}}

$$

Dirac spinor corresponds to the direct sum $\left( \frac{1}{2}, 0\right) \oplus \left( 0, \frac{1}{2}\right)$:

$$

\Psi = \begin{pmatrix} \psi_{a} \\ \kappa^{\dot{b}}\end{pmatrix}

$$

Charge conjugation of Dirac spinor is determined as

$$

\tag 1 \Psi \to \hat{C}\Psi = \begin{pmatrix} \kappa_{a} \\ \psi^{\dot{b}}\end{pmatrix}

$$

Sometimes charge conjugation is called "the Dirac version of of complex conjugation".

The question. Why do we need to introduce complex conjugation in a form $(1)$, not in a form of ordinary complex conjugation, $\Psi \to \Psi^{*}$?