No, carbon cannot undergo Ionic bonding. For carbon to be able to go through this bonding, they would have to lose or find four electrons. This would cause the carbon to become unstable, and therefore it is impossible.
The four valence electrons would need to be able to form and bond single, double and triple strands, as each gas for each particle has to be accounted for. Even though ionic bonds cannot be formed, there can be some different bonds forms.
As carbon is among the highest used element on the periodic table, there are many other bonding and processes that it can go through. Covalent bonds are an example of one. There are different forms of these bonds, and it can have up to four different types of relationships as well.