Analytic continuation of Riemann zeta $zeta(s)$ from the complex $mathbb{C}$ to quaternion $mathbb{H}$?

One way to define Riemann zeta function is by the analytic continuation of

$$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$
for the domain $Re(s)>1$ to the full complex plane in $\mathbb{C}$.

Thus, Riemann zeta function is defined for $s \in \mathbb{C}$ and $\zeta(s) \in \mathbb{C}$

My question is that do we gain anything new to do analytic continuation of Riemann zeta function such that a "modified Riemann zeta function" so

$s \in \mathbb{H}$ is in quaternion? and $\zeta(s) \in \mathbb{H}?$

Does this lead to any interesting result in the math literature?

Edit: more precisely, according to the comment, we seek for an analytic continuation of $\zeta(s)$ from the complex $\mathbb{C}/\{1\}$ to quaternion $\mathbb{H}/\{1\}$?