Key idea. Use C2.3 and ask when one of the integer factors can equal 1.

1. The factorisation is

   \[x^4+4y^4=\bigl((x-y)^2+y^2\bigr)\bigl((x+y)^2+y^2\bigr).\]

2. For positive integers x and y, both factors are positive integers. The second is at least \(2^2+1^2=5\).

3. If their product is prime, the first factor must therefore be 1:

   \[(x-y)^2+y^2=1.\]

   Since \(y\ge1\), we must have \(y=1\) and \(x-y=0\), giving \(x=1\).

4. Conversely, \((x,y)=(1,1)\) gives \(1+4=5\), which is prime. No other positive integer pair can work.

Remember. A prime written as a product of positive integers forces all but one factor to be 1.