As long as the form is positive definite and the unit ball is convex, you get a perfectly good Banach space using any symmetric $n$-linear form on a real vector space $V$.  The degree $n$ is necessarily even.  It is equivalent to defining the norm as the $n$th root of a homogeneous degree $n$ polynomial.  $\ell^p$ is an example for any even integer $p$.   There are many other examples.  I found a paper, [Banach spaces with polynomial norms][1], by Bruce Reznick, that studies these norms.  He obtains various results; the most appealing one to me at a glance is that these Banach spaces are all reflexive.

Off-hand I can't think of any simple way to recover positive definiteness starting with odd polynomials.  The cube of the norm on $\ell^3$ is a polynomial in the absolute values of the coordinates rather than the coordinates themselves.

Addendum: To address Darsh's comment, what you would look at in the complex case is self-conjugate polynomials of degree $(n,n)$.  Equivalently, as with all complex Banach norms, the realification is a real Banach norm which is invariant under complex scalar rotation.

  [1]: https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-82/issue-1/Banach-spaces-with-polynomial-norms/pjm/1102785074.full