I don't actually understand what he is proving, nor how he claims to prove it.
I've now found and read the discussion on slashdot (here) - I didn't find that before because I searched on the guy's name which somehow escapes mention in the entirety of the discussion there.
The main questions are:
1. does this give an algebraic solution or a numerical solution? That is, does it give a formula to express the roots of an arbitrary polynomial in terms of elementary functions, or does it merely give an algorithm that allows you to approximate the result? There is a big difference between knowing that sqrt(2) is the answer and finding that it is about 1.41421356...
An algebraic solution would contradict Abel's impossibility theorem, generally believed to have been "rigorously proved"; such a contradiction would call into question a whole lot of group theory (apart from anything else), possibly to the extent of bringing into question most of the last 50 years of theoretical physics research (much of which revolves around symmetry groups).
The slashdot discussion is unsurprisingly not particularly clear, but there seems to be a fairly good consensus that the proof is actually presenting a numerical approach, and it is the Dutch-language media that have misinterpreted the results as being an algebraic solution.
If it is a numerical solution, Abel's theorem is not called into question, and maths survives another day.
2. If it is a numerical method, is it useful?
There already exist many different approaches for finding numerical solutions to polynomials. Comparing such approaches involves looking at applicability, complexity and convergence.
The claims of this paper score high on applicability: it claims to be applicable for every polynomial, which means using this approach a) you don't need to do extra work up front to determine whether your polynomial is solvable using this method; and b) you don't need to implement one or more fallback methods to use in the cases where it isn't applicable.
The complexity determines how much work you need to do for each step of the iteration, and the convergence determines how many iterations you need to perform to calculate roots to a given level of accuracy. I have no idea how this approach rates compared to any other approach.
3. Forget all that, is it correct? And is it new?
No idea. Slashdot isn't certain either.