Let $M$ be a real-analytic manifold and let $N$ be a complexification of $M$ (in other words, $M$ sits in $N$ as a totally real submanifold). Suppose $M$ has an (integrable) complex structure. Does it extend in a neighborhood of $M$ in $N$ (in a unique "natural" way?) such that $N$ becomes a manifold with a quaternionic structure? (If not, what additional data/restrictions are required?)
What if $M$ has a real-analytic Kahler structure? Does it extend in N as a hyperkahler structure? If so, what does this have to do with the hyperkahler structure in (a neighborhood of the zero section in) the holomorphic cotangent bundle of $M$ described by Feix and Kaledin?
Thanks in advance for your insights.