If there are any terms here you don't understand, you really should read the linked writeups. The really important ones are complex numbers and Gaussian Integers.
A Gaussian Mersenne Prime number is defined as follows:

The complex numbers, normally called the set C, have a subset, the Gaussian Integers, Z. These are numbers of the form A+Bi, where A and B are integers. Since a Mersenne number is an integer of the form 2^{n}-1, similarly, a Gaussian Mersenne number is a complex number of the form (1±i)^{n}-1, that is also a Gaussian Prime (this is very different than what the E2 w/u has as a Gaussian Prime, it may be that there are 2 seperate definitions.)

A Gaussian integer a+bi is a Gaussian prime if and only if its norm N(a + bi) = a^{2}+b^{2} is prime or b=0 and a is a prime congruent to 3 (mod 4). For example, the prime factors of two are 1+i and 1-i, both of which have norm 2. So we have the following:
(1 ± i)^{n} – 1 is Gaussian Mersenne prime if and only if n is 2, or n is odd and the norm 2^{n}-(-1)^{((n^2)-1)/8}2^{(n+1)/2}+1 is a rational prime.

In addition, the case where, instead of having (1 ± i)^{n} – 1, we have (1 ± 1)^{n} – 1 (because there is really no reason to limit it to an imaginary additive to the original 1.) This yeilds two additional cases: -1, which is a trivial case; (1-1)^{n}-1 is -1, or the normal case of Mersenne primes, 2^{n} – 1.

Gaussian Mersenne Primes, while not having been researched very fully, have been proven to share a number of interesting characteristics with perfect numbers and with regular Mersenne Primes. It has also been suggested that they occur with the same density as Mersenne Primes, but this remains an open question. They are indexed by the exponent, and there are a total of 35 known (exponent listed); 2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289. It is sequence A057429 on the On-Line Encyclopedia of Integer Sequences.

Sources:

http://primes.utm.edu/glossary/page.php?sort=GaussianMersenne

http://www.mail-archive.com/mersenne@base.com/msg05162.html

http://mathworld.wolfram.com

http://www.research.att.com/~njas/sequences/