8
A+B+C=0
A=-B-C
B^2+C^2+BC=2
B^4+2B^3C+3B^2C^2+2BC^3+C^4=4
A^4+B^4+C^4=2(B^4+2B^3C+3B^2C^2+2BC^3+C^4)
=2*4=8