Topological K-theory is usually defined by setting $K(X)$ to be the groupification of the monoid $Vect_\mathbb{C}(X)$ of complex vector bundles over $X$ (with addition given by Whitney sum). However, we can alternatively declare that $[B]\sim [A]+[C]$ whenever $0\rightarrow A \rightarrow B\rightarrow C\rightarrow 0$ is a short exact sequence of vector bundles over $X$ (morphisms are required to have locally constant rank): certainly if $B\cong A\oplus C$ then we have such a sequence, and in the other direction we can take a metric on $B$ and identify $C$ with $A^\perp \subseteq B$.

You can take the $K$-theory of any abelian category using this second definition. So, I'm curious to know if people do this for the category of holomorphic vector bundles over a complex manifold. The above splitting construction no longer works since it uses partitions of unity, so assuming we use this more general definition we'd get more equivalence relations. On the other hand, there's all this funny business going on with vector bundles topologically but not holomorphically isomorphic, which means that $K_{hol}(X)$ wouldn't just be a subquotient of $K(X)$. So in the end, I'm not sure whether I should expect this to be a more or less tractable sort of object.

I'm told that the Chow ring might have something to do with this, but the wikipedia page seems to indicate that it's more analogous to singular cohomology than anything else.