  We show that the variety of monadic ortholattices is closed under MacNeille
and canonical completions. In each case, the completion of $L$ is obtained by
forming an associated dual space $X$ that is a monadic orthoframe. This is a
set with an orthogonality relation and an additional binary relation satisfying
certain conditions. For the MacNeille completion, $X$ is formed from the
non-zero elements of $L$, and for the canonical completion, $X$ is formed from
the proper filters of $L$. The corresponding completion of $L$ is then obtained
as the ortholattice of bi-orthogonally closed subsets of $X$ with an additional
operation defined through the binary relation of $X$.
  With the introduction of a suitable topology on an orthoframe, as was done by
Goldblatt and Bimb\'o, we obtain a dual adjunction between the categories of
monadic ortholattices and monadic orthospaces. A restriction of this dual
adjunction provides a dual equivalence.
