If w is real and z complex, this is the root-locus problem of control theory. As noted by other contributors, there is no way to solve a sixth-degree polynomial equation in closed form, so obtaining exact (analytic) solutions for z=z(w) is a challenge. Depending on what you need z(w) for, some of the following devices might be of help.
Although Maple's plots package contains a rootlocus command, it numerically solves and graphs solutions for 1+k*h(z), where k is a real parameter (called a "gain" in the language of controls).
If z is real, then a graph of z=z(w) can be obtained by solving for w=w(z), then graphing via plot([w(z),z,z=a..b]).
If z is complex, solve for w=w(z) and replace z with x+I*y. Let Wr be the real part and Wi be the imaginary part. (This takes evalc(Re(... and evalc(Im(... but for polynomials, Maple can do this.) Now, if w is real, then Wi is zero, so an implicitplot of Wi=0 is a graph of the root locus. Each point on the resulting graph is a z=x+I*y value in the xy-plane for which some w satisfies the original equation P=0. The trick now is to obtain that w-value for each point on the curve.
For example, pick an x-value and count how many y-values will correspond. Then set x in Wr equal to that value and fsolve for y. Make a procedure of this process and you will be able to see the values of w for each z on the root locus. In control theory, a value of z indicates a possible state of a system. The idea is to determine the gain that will put the system into that state, so knowing the gain as a function of the state is a useful thing, something not ordinarily discussed in elementary texts in the subject.
As an isolated mathematical problem, I have always found this topic interesting. Just what is the trajectory in the z-plane of the solutions of P=0 as w varies, even if w isn't real. It's a hard problem to solve directly, so looking at it "backwards" certainly leads to interesting options.