a^4+4 is a special case of a^4+4b^2, with b=1.

Factoring a^4+4b^2 = (a^2+2ab+2b)*(a^2-2ab+2b).

Substitute b=1 to get (a^2+2a+2)*(a^2-2a+2). The first term is greater than 1 for any a!=-1. For the product to be prime one of the factors must equal 1.

For any integer a greater than 1 or less than -1, both terms are positive integers greater than 1, which makes the product composite.

This leaves a=-1, 0, 1 as potential prime generators If a=1 or -1 then a^4+4=5 (prime), and if a=0 then a^4+4=4 (composite).

Then the set of all integer values a which make a^4+4 prime is {-1,1}.