This number is a prime.
The largest known Gaussian Mersenne prime is (1+i)^1203793-1 (found September 2006 by B. Jaworski). The real and imaginary parts of this behemoth both have 180989 digits and its norm (the square root of its absolute value) is a 362378-digit prime. There are 36 other Gaussian Mersenne primes known. [Caldwell]