What is a polynomial? A better way to visualize them

What I’m try to convey is that the usual visualization of polynomials as a plot in Cartesian coordinates is somewhat limited, especially when it comes with higher dimensions. I hope there is some other kind of visualization that one can use (even if it’s not very accurate or rigorous) drawing from other fields (e.g. Galois theory, differential geometry, physics or even something not directly related to mathematics). For example one can think about roots as ground states of the polynomial.

What best describes the intention of this question is this minutephysics video https://www.youtube.com/watch?v=W2Xb2GFK2yc

I’d like to know if there is a non conventional way to picture in one’s mind the concept of a polynomial.