What are the polynomial generalizations of
Eisenstein-Mersenne Primes
Primes of the form $3^n+3^{(n+1)/2}+1$ if $n$ $=$ $5, 7$ $\pmod {12}$ and $3^n-3^{(n+1)/2}+1$ if $n$ $=$ $1, 11$ $\pmod {12}$.
Gaussian-Mersenne Primes
Primes of the form $2^n+2^{(n+1)/2}+1$ if $n$ $=$ $3, 5$ $\pmod 8$ and $2^n-2^{(n+1)/2}+1$ if $n$ $=$ $1, 7$ $\pmod 8$.
Both these conditions are the same as the Mersenne Prime Conditions, in order for the expressions above to be prime, $n$ must also be prime. Also, if $n$ is prime, all factors all the Gaussian or Einstein Mersenne Prime must be of the form $kn+1$.
Question $1$: Is there are generalized polynomial form (of the form $b^n±b^k+1$) for Gaussian and Eisenstein Mersenne Primes which holds the same conditions listed as the original Mersenne Primes?
Assuming the answer is yes, for question $1$, what are the generalized forms for the Gaussian (base $2$) and Eisenstein (base $3$) primes, for base $b$, of the form $b^n±b^k+1$ where it is $-b^k$ if $b$ is a quadratic nonresidue $\pmod n$ and $+b^k$ if $b$ is a quadratic residue $\pmod n$.
See https://en.wikipedia.org/wiki/Mersenne_prime#Generalizations for more information. Thanks for your help.