Lets define the power symmetric group
$$\text{psg}_{n,d}(x_1,\dots, x_n) = \sum_{i=1}^n x_i^d$$
And lets define a linear form as
$$\ell_i(x_1,\dots,x_n)=\sum_{j=1}^{n} a_{ij}x_i+a_{i0}$$
Where $a_{ij} \in \mathbb{C}$
For convenience, I'll drop the variables from the functions. I am searching for two sets of linear forms $\{\ell_i\}_{i \in [n]}$ and $\{p_i\}_{i \in [n]}$ where
$$\sum_{i=1}^{n} \ell_i^d = \sum_{i=1}^{n} p_i^d=psg_{n,d}$$
But there doesn't exist integers $e_1,\dots, e_n$ and a permutation $\pi \in S_n$ where $\forall i \in [n]~ p_i = \omega^{e_i} \cdot\ell_{\pi(i)}$ where $\omega$ is a primitive $d$-th root of unity.
According to the paper Affine projections of polynomials by Neeraj Kayal that was published in $2011$, this claim is true but I could not find an example of one myself.
It seems like a simple example of this is $$x^2 + y^2 = \left(\frac45 x + \frac35y\right)^2 + \left(\frac35 x - \frac45y\right)^2.$$ More generally, when $d=2$, if we take any $n \times n$ matrix $A$ such that $A^{\mathsf T}\!A = I$ (any orthogonal matrix), then $A \mathbf x$ gives us $n$ linear forms that will do the trick.