Examples of context-free languages with a non-context-free complements
The language $L_1= \{ww \mid w \in \{a,b\}^*\}$ is not context-free (as can be shown using the pumping lemma; see here).  Its complement $L_2 = \{a,b\}^* \setminus L_1$ is context-free (as shown here).  This gives a simple and elegant example of a context-free language (over a binary alphabet) whose complement is not context-free, as you requested.