Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
Just as straightedge and compass constructions give the numbers in the closure of the rationals under square roots, neusis gives the closure of the rationals under square roots and cube roots.
For more details, also for an alternate characterization in terms of origami, see this paper by Roger Alperin.
There are various flavors of neusis construction. In the weakest flavor, in addition to having the marked straightedge pass through the pole point, the two marks on it are required to lie, one each, on two specified lines; we might call that tool a line-line neusis. For a line-circle neusis, one mark must lie on a specified line while the other must lie on a specified circle. For a circle-circle neusis, the two marks must lie, one each, on two specified circles. (We view a line as a special case of a circle; so each of these tools is at least as powerful as its predecessors.)
If we allow ourselves a straightedge, a compass, and a line-line neusis, then, as Stillwell tells us, we get precisely the closure of the rationals under complex square roots and cube roots. The Alperin paper that Stillwell mentions is a high-level reference. Here are some more details.
In one direction, consider the line through the pole point that has slope $s$. We can intersect that line with the two specified lines. The distance between the two resulting intersections equals the fixed distance between the two marks on the straightedge just when a certain quartic equation in $s$ holds. And, of course, any quartic can be solved using complex square roots and cube roots.
In the other direction, the compass allows us to bisect any angle and to extract any real square root; so we can take complex square roots. To show that the line-line neusis can take complex cube roots, we need to show two things: that it can trisect any angle and that it can extract any real cube root.
Trisecting first: There is a well-known neusis angle-trisection credited to Archimedes; but that construction uses a line-circle neusis, and hence doesn't help us here. But the Greeks also knew of a trisection using a line-line neusis. Alperin credits that construction to Apollonius, but gives no details. For the details, see either A History of Greek Mathematics, Volume 1: From Thales to Euclid, by Sir Thomas Heath, reprinted by Dover in 1981, pages 236-238. Or see Exercise 10 on page 245 of Michael O'Leary's Revolutions in Geometry, published by Wiley in 2010. (Note that, in this construction, of the four slopes for the neusis straightedge that satisfy the distance requirement, all four are real; one is trivial and should be ignored, while the other three are the three trisectors.)
Now for real cube roots: The Greeks also knew a line-line neusis construction, credited to Nicomedes, for extracting real cube roots. One source for that construction, pointed out by Gerry Myerson, is the article "Constructions using a compass and twice-notched sraightedge", by Arthur Baragar, pages 151-164 in volume 109, number 2 of the American Mathematical Monthly. (In this construction, of the four slopes mentioned above, one is trivial and should be ignored, a second gives the required real cube root, and the remaining two are complex.)
One side remark: The Nicomedes cube-root construction is a bit subtle. But Conway and Guy give a dead-simple line-line neusis construction for the special case of the cube root of 2, on page 195 of The Book of Numbers, published by Springer in 1996.
So the line-line neusis gives us precisely the power to solve quadratic and cubic (and hence also quartic) equations, which makes it equivalent to "conic constructability" or, as the Greeks called it, "solid constructability".
What about the other flavors of neusis construction? Baragar shows that, for either a line-circle neusis or a circle-circle neusis, there are in general six slopes for the line through the pole that have the proper distance relationship -- six, rather than the four of the line-line case. He then gives an explicit example of a line-circle neusis construction in which one of these slopes is real and trivial, three others are real, and the final two are complex. Furthermore, the five nontrivial slopes are the roots of a irreducible quintic equation whose Galois group is all of $S_5$, and which hence cannot be solved with radicals. Thus, the line-circle neusis is a strictly more powerful tool than the line-line neusis.
As an upper bound on the power of these more general neusis constructions, Baragar shows that any point generated by either the line-circle neusis or the circle-circle neusis lies in an extension field of the rationals that can be reached by a tower of fields in which each adjacent pair has index either 2, 3, 5, or 6. So the jump in power over the line-line neusis, where the adjacent-pair indices are either 2 or 3, is not too great.
Baragar's paper closes with some interesting open problems. One problem that he doesn't mention is this: Is the circle-circle neusis strictly more powerful than the line-circle neusis?