Step 1: Use the identity for root differences:
For roots $\alpha, \beta$: $(\alpha-\beta)^2 = (\alpha+\beta)^2 - 4\alpha\beta$.
Here $\alpha+\beta = -2$ and $\alpha\beta = 4$.
Step 2: Compute:
$(\alpha-\beta)^2 = 4 - 16 = -12$.
So $\alpha - \beta = \sqrt{-12} = 2\sqrt{3}\,i$ (taking the positive imaginary root as the difference).
Step 3: Check against the options:
Only option (C) equals $2i\sqrt{3}$. The real options (A) and (B) would need a positive discriminant, and $i\sqrt{3}$ in (D) is half the difference.
Final Answer:
$\alpha-\beta = 2i\sqrt{3}$, option (C).
\[ \boxed{2i\sqrt{3} \text{ (C)}} \]