  We study $2\times 2$ matrices over noncommutative rings with anti-involution,
with a special focus on the symplectic group
$\mathrm{Sp}_2(\mathcal{A},\sigma)$. We define traces and determinants of such
matrices and use them to prove a Cayley Hamilton identity and trace relations
which generalize well known relations for elements of $\mathrm{SL}_2(R)$ over a
commutative ring. We compare the structure of elements of
$\mathrm{Sp}_2(\mathcal{A},\sigma)$ with Manin matrices over general
noncommutative rings; this naturally leads to a quantization
$\mathrm{Sp}_2(\mathcal{A},\sigma)_q$. In contrast to the usual definition of
the quantum group as a deformation of the ring of matrix functions on
$\mathrm{SL}_2(R)$, this quantization produces a group of matrices over a new
noncommutative ring with involution. We finish the comparison by constructing a
generalization of a Hopf algebra structure on the noncommutative ring of matrix
functions of our quantum group. Finally, we use the noncommutative surface-type
cluster algebras of Berenstein and Retakh to give a geometric interpretation of
our Hopf algebra structure and to produce noncommutative generalizations of
Markov numbers over many rings with involution including the complex numbers,
dual numbers, matrix rings, and group rings.
