Let $\mathbb{H} \subset M_2(\mathbb{C})$ be the set of matrices of the form:
$$A = \begin{pmatrix}z & -\bar{w}\\w & \bar{z}\end{pmatrix} \text{ where } z,w \in \mathbb{C}$$
$A^{-1} = \begin{pmatrix}z & -\bar{w}\\w & \bar{z}\end{pmatrix}^{-1} = \frac{1}{(\det A)} \begin{pmatrix}\bar{z} & \bar{w}\\-w & {z}\end{pmatrix} = \frac{1}{(\bar{z}z + w\bar{w})} \begin{pmatrix}\bar{z} & \bar{w}\\-w & z\end{pmatrix} = A^{-1}$
So, ${AA^{-1} = I \qquad\forall A \in \mathbb{H}^\ast = \mathbb{H}\setminus\{0\}}$
Hence, $\mathbb{H}$ is a division ring
It is correct, except that you should add to it that $z\overline z+w\overline w\in\mathbb R$. If $\lambda$ is an arbitrary complex number, it is not true that $\lambda\left(\begin{smallmatrix}z&-\overline w\\w&\overline z\end{smallmatrix}\right)\in\mathbb H$. And, instead of “$=A^{-1}$”, you should have written “$\in\mathbb H$”.