Infinitely many solutions to $a^4+b^4+c^4=18$ over $\mathbb{Z}[i]$ We got infinitely many solutions to $a^4+b^4+c^4=18$ over 
$\mathbb{Z}[i],i^2=-1$. Probably we can get infinitely many solutions
to $a^5+b^5+c^5=N$ over $\mathbb{Z}[\alpha]$ for algebraic $\alpha$.

Are these results known and/or trivial?

There is some chance an improvement of the method to solve similar
equations over the integers.
 A: Let $K$ be a number field. Then the Bombieri–Lang–Vojta–... conjecture asserts that for $d \geq 5$, all but finitely many of the $K$-points of the smooth projective surface of general type
$$X = V(x_0^d+x_1^d+x_2^d-Nx_3^d) \subseteq \mathbb P^3$$
are contained in some strict subvariety $Z \subsetneq X$. A stronger version asserts that we can choose $Z$ independent of $K$, and that we may take it to be the union of all rational and elliptic curves in $X$. This is a good thing to try.
Over $\mathbb C$ and if $d$ is odd, we have a map
\begin{align*}
\mathbb P^1 &\to X\\
[x:y] &\mapsto [x:-x:y\sqrt[d]{N}:y].
\end{align*}
There are obvious modifications one can make for $d$ even. This exhibits a specific $\mathbb P^1$ on $X$, which is actually defined over $\mathbb Q[\sqrt[d]{N}]$.
This already gives infinitely many $\mathbb Q[\sqrt[d]{N}]$-points on $X$. Setting $y = 1$ and varying $x$ over $\mathbb Z[\sqrt[d]{N}]$, we actually get infinitely many integral points. Explicitly, I have exhibited for you the points
$$(a,b,c) = (x,-x,\sqrt[d]{N}),$$
where $x \in \mathbb Z[\sqrt[d]{N}]$ is arbitrary.
One could probably have found these without referring to fancy conjectures. On the other hand, the conjecture can serve as a guide: conjecturally, this type of construction is the only way in which we can produce infinitely many rational points. In general, however, trying to produce maps $\mathbb P^1 \to X$ is probably not your winning strategy for producing rational or integral points...
