Let $\mathbb{H}$ be the skew-field of quaternions. I'm aware of the
Theorem 1. A function $f:\mathbb{H}\to\mathbb{H}$ which is $\mathbb{H}$-differentiable on the left (i.e. the usual limit $h^{-1}\cdot (f(x+h)-f(x))$, for $h\to 0$, exists for every $x\in \mathbb{H}$) is a quaternionic affine function on the right (i.e. of the form $x\mapsto x\cdot \alpha + v$).
This means that there are no interesting smooth quaternionic funcions, hence no interesting "quaternionic-smooth manifolds" (which is not the same as the quaternionic-Kahler or hyperkahler structures you encounter in differential geometry and complex analytic geometry).
I think I can also recall the
Theorem 2. If a function $f:\mathbb{H}\to\mathbb{H}$ is locally $\mathbb{H}$-analytic (i.e. it can be locally developped in power series, for the suitable noncommutative notion of "power series"), than it corresponds to a real-analytic function $f:\mathbb{R}^4\to \mathbb{R}^4$, and any real-analytic funcion $f:\mathbb{R}^4\to \mathbb{R}^4$ can be obtained in this way.
That says that $\mathbb{H}$-analytic functions are essentially the same as quadruples of real-analytic functions of 4 variables. Hence there is no "quaternionic-analytic geometry" distinguishable from $4n$-dimentional real-analytic geometry.
I think the same happens with quaternionic (noncommutative) polynomials: they're just 4-tuples of real polynomials in 4 variables.
But, is it reasonable that the zero locus on $\mathbb{H}^n$ of a "noncommutative polynomial" with $\mathbb{H}$-coefficients doesn't have any further mathematical structure than it's real-algebraic variety structure?
It would be nice to be able to see things such as $\mathbb{HP}^1$ as a "quaternionic curve ", and to speak of a point " $\mathrm{Spec}(\mathbb{H})$ " (whatever it means) if it possible…
Is there a theory of "quaternionic algebraic geometry", maybe as a branch or particular case of some noncommutative (algebraic) geometry theory?
Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show.
Best Answer
The answer is yes!(at least if quaternionic holomorphic geometry counts) "Qauternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space.
The first paper in this field is more or less
A good introduction to "Quaternionic holomorphic geometry" is given by
and
But just to clarify things: "Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show."
This isn't really true, because the quaternionic holomorphic geometry developed in the papers above, is a kind of generalization of complex geometry, i.e. if you look how a "quaternionic holomorphic structure" is defined, you see that it is a kind of $\overline{\partial}$-Operator with some extra data (a so called Hopf field).
If you need more references, there are plenty available (just type "quaternionic holomorphic geometry" into google)