I am thinking on the complex analogue of the Mersenne primes.
I think, some like a "complex Mersenne prime" could be a complex prime in the form
$$2^{a+b\frac{pi}{2}i}-1$$
Where $a+bi$ is a complex prime as well.
Is it an "usable" extension in the sense, that its prime testing could be more fast as the "normal" complex primes? What I practically need, were big complex primes whose primeness was tested without random number generation.
Your idea of extending the concept of Mersenne primes to complex exponents seems like purely wishful thinking, as Micah points out above. If you want large Gaussian primes whose primality is somewhat easier to test, I would consider Proth primes, which are numbers of the form $k\cdot 2^n+1$. If you restrict $k$ to be a perfect square $>1$ and $n$ to be even, then finding that $k\cdot 2^n+1$ is prime gives you the Gaussian prime $\sqrt k \cdot 2^{n/2} + i$.
Pretty large example from http://www.prothsearch.net/riesel.html: $9\cdot 2^{3497442}+1$ is prime.