we develop a new approach to gelfand-zeitlin theory for the symplectic group $sp(n,\mathbb{c})$. classical gelfand-zeitlin theory, concerning $gl(n,\mathbb{c})$, rests on the fact that branching from $gl(n,\mathbb{c})$ to $gl(n-1,\mathbb{c})$ is multiplicity-free. since branching from $sp(n,\mathbb{c})$ to $sp(n-1,\mathbb{c})$ is not multiplicity-free, the theory cannot be directly applied to this case. let $l$ be the $n$-fold product of $sl(2,\mathbb{c})$. our main theorem asserts that each multiplicity space that arises in the restriction of an irreducible representation of $sp(n,\mathbb{c})$ to $sp(n-1,\mathbb{c}$, has a unique irreducible $l$-action satisfying certain naturality conditions. we also given an explicit description of the $l$-module structure of each multiplicity space. as an application we obtain a gelfand-zeitlin type basis for the irreducible representations of $sp(n,\mathbb{c})$.