  We give a new interpretation of representation theory of the
finite-dimensional half-integer weight modules over the queer Lie superalgebra
$\mathfrak{q}(n)$. It is given in terms of Brundan's work of finite-dimensional
integer weight $\mathfrak{q}(n)$-modules by means of Lusztig's canonical basis.
Using this viewpoint we compute the characters of the finite-dimensional
half-integer weight irreducible modules. For a large class of irreducible
modules whose highest weights are of special types (i.e., totally connected or
totally disconnected) we derive closed-form character formulas that are
reminiscent of Kac-Wakimoto character formula for classical Lie superalgebras.
