Here's what I have in mind: for $k=\Bbb C$ algebraic equations with $k$-coefficients lead to a $k$-variety in $\Bbb A_k^n$, and that variety inherits the structure of $k$-manifold.

However, the above is not necessarily true when $k=\Bbb H$ is the skew-field of quaternions: the variety doesn't have to have $dim_{\Bbb R}$ divisible by $dim_{\Bbb R}\Bbb H$, let alone a quaternionic structure.

Therefore the question: what properties of (ring, skew-field, ...) $k$ cause a $\Bbb A_k$-variety have a natural manifold structure associate with $k$, whatever it is?