The index of $\xi_4^*$ in $\xi_4$ Just seeing if i'm right:  
With the set of solutions for $z^4=1$: $\xi_4=\{1,i,-1,-i\}$, one can construct the group of the $4$th roots of unity: $(\xi_4,\cdot_\mathbb{C})$ and its multiplicative subgroup $(\xi_4^*,\cdot_\mathbb{C})$ with $\xi_4^*=\{1,-1\}$.  

What is the Index $[ \xi_4 : \xi_4^* ]$?  

I'd say through Lagrange:
$$ [ \xi_4 : \xi_4^* ] =\frac{|\xi_4|}{|\xi_4^*|}=\frac{4}{2}=2.$$
Or counting the cosets of $\xi_4^*$: 
$$1\cdot\xi_4^*=\{1,-1\}= -1\cdot\xi_4^*,\quad i\cdot\xi_4^*=\{i,-i\}= -i\cdot\xi_4^*$$
we get two.
Is this correct?

Yes, indeed, the index $$[\xi_4:\xi^*_4] = \frac{|\xi_4|}{|\xi^*_4|} = \dfrac 4 2 = 2$$ 
Alternatively, as you've shown, the index $=$ the number of cosets of $\;\xi^*_4 \leq \xi_4 = 2$.
REMARK:
When you're dealing with a finite group, as in this case, the first "method" of computing the index of a subgroup with respect to its containing group is probably quickest. (Indeed, the first method works strictly for finite groups!)
