OP has a point. In eq. (28.22) the matrix-valued field $\Sigma\in{\rm Mat}_{2\times 2}(\mathbb{C})$ has 8 real DOF, so $\sigma(x),\pi^a(x)\in\mathbb{C}$ in eq. (28.24) are in principle complex fields. After SSB $$G~=~SU(2)_L\times SU(2)_R\quad\longrightarrow\quad H~=~SU(2)_D,$$ the 3 real parts ${\rm Re}\pi^a(x)$ becomes massless Goldstone modes.
OP has a point. If the matrix-valued field $\Sigma\in{\rm Mat}_{2\times 2}(\mathbb{C})$ in eq. (28.22) has originally 4 complex DOF, then $\sigma(x),\pi^a(x)\in\mathbb{C}$ in eq. (28.24) are in principle 4 complex fields. Then after SSB $$G~=~SU(2)_L\times SU(2)_R\quad\longrightarrow\quad H~=~SU(2)_D,$$ the 3 real parts ${\rm Re}\pi^a(x)$ becomes massless Goldstone modes.
Note however that to make sense of the text after eq. (28.26) it seems that the matrix-valued field $\Sigma\in \mathbb{R}_+\times SU(2)$ in eq. (28.22) has originally 4 real DOF. Then $\sigma(x),\pi^a(x)\in\mathbb{R}$ in eq. (28.24) are 4 real fields, so that the counting of DOF is OK.