12. Given $a-b=1, a^{2}-b^{2}=-1$. Then $a^{2008}-b^{2008}=$
12. -1 .

Given $a^{2}-b^{2}=(a+b)(a-b)=-1$, and $a-b=1$, then $a+b=-1$.
Therefore, $\left\{\begin{array}{l}a+b=-1 \\ a-b=1 .\end{array}\right.$ Solving, we get $\left\{\begin{array}{l}a=0, \\ b=-1 .\end{array}\right.$ Hence, $a^{2 \alpha R}-b^{20 R}=0^{2008}-(-1)^{2008}=-1$.