Suppose we have a complete lattice. Which additional axioms (e.g. distributivity axioms) are needed to obtain a Boolean lattice in which complement(a) = lub{b  b /\ a = bottom} = glb{b  b / a = top}?

Since you are asking about universally quantified equations, there is only the trivial answer: $x=y$. Theorem: Assume that $\varphi$ is a (finite or infinite) list of universally quantified equations (sometimes called "laws" or "identities") such that every complete lattice satisfying $\varphi$ will have complements. Then there is no nontrivial lattice satisfying $\varphi$, or in other words, the equations $\varphi$ imply $(\forall x,y): x=y$. Proof: From a nontrivial lattice satisfying $\varphi$ I will construct a complete lattice satisfying $\varphi$ which is not Boolean. All the following lattices will satisfy $\varphi$: (a) Some nontrivial lattice with at least 2 points $p < q$. (b) The 2point Boolean lattice containing only $p < q$. (A sublattice of (a)). (c) $\{p,q\}^2$, a 4element Boolean lattice. (d) $\{(p,p), (p,q), (q,q)\}$, a sublattice of (c). This is a 3element chain. The lattice in (d) is complete, satisfies $\varphi$, but is not Boolean. 

