For a complex manifold $X$ one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain $K$-theory (I do not know whether the $K$-groups given by the "standard" constructions are isomorphic in this setting) and $\Lambda$-operations for them.

My question is: did anyone study these $K$-groups and $\gamma$-filtration for them; are there any methods to compute them (at least, for some examples) if $X$ is compact but not algebraic?