Complicity = involvement in a crime or some activity that is wrong.

• Police suspected his complicity in Billa’s escape.
• The police are investigating the extent of the manager’s complicity in the fraud.
• They are accused of complicity in the murder case.
• His complicity in the bank robbery is the centre point of police investigation.

Complexity = the state of being complex or complicated.

• His dismissal is an issue of great complexity.
• The complexities of starting an exporting business.
• Don’t get involved in the project without examining the probable complexities that might be remaining hidden.
• The position becomes more complex where an investigation is underway.