Consider how one would numerically generate equipotentials, given the
conservative vector field E(x,y):
1) Beginning at some chosen point (x,y), proceed an amount ds in a
direction perpendicular to the direction of the local E field.
2) Continue in this manner, tracing out your path in x,y space.
3) This generates an equipotential line.
4) Choose a new starting point and repeat this procedure to generate a
different equipotential, etc.
Now, one can also perform this procedure using a NON-conservative field
E(x,y). The important difference is that the paths so generated are not
equipotentials (No potential function exists). I would not even call them
pseudo-equipotentials. They are, by construction, paths along which a
charge can move without exchanging any energy with the E field. They
might be called paths of constant energy, although the value of that
energy (for a given path) is not unique, even as measured from some chosen
reference point. A path may not be labeled with an energy value ( as an
equipotential can), the energy value depends on the detailed history of
how you got to this path, from the reference point. This seems to be as
close as one might get to an analogy of the equipotential concept of a
conservative E field.
For John's betatron field these paths of constant energy would be radial
lines. They may serve some useful pictorial purpose, but I fear they
invite easy misinterpretation.