  D=4, N=4 super-Yang-Mills theory has an off-shell superspace formulation in
terms of pure spinor superfields, which is directly inherited from the D=10
theory. That superspace, in particular the choice of pure spinor variables, is
less suitable for dealing with fields that are inherently 4-dimensional, such
as the superfields based on the scalars, which are gauge-covariant, and traces
of powers of scalars, which are gauge-invariant. We give a reformulation of
D=4, N=4 super-Yang-Mills theory in N=4 superspace, using inherently
4-dimensional pure spinors. All local degrees of freedom reside in a superfield
based on the physical scalars. The formalism should be suited for calculations
of correlators of traces of scalar superfields.
