Chairman Bob Dench has begun the process to find Bester's successor, the bank said.
Bester - who joined the bank in July 2018 - has committed to remain in place until that process is complete to ensure a smooth and orderly transition. The effective date of his departure will be confirmed in due course.
In August, the bank said it would cut around 350 jobs and close 18 branches as the coronavirus crisis forced it to reduce costs.
The Co-Operative Bank is one of several British challenger lenders attempting to challenge the dominance of the Big Four high-street banks including Lloyds Banking Group, HSBC, NatWest Group and Barclays.
But like others, it has found its progress hampered by low interest rates, blistering mortgage competition and erratic consumer demand for loans as Britain's economy is rocked by the impact of the COVID-19 pandemic.
It now joins larger rival Lloyds in the search for a new leader after its CEO Antonio Horta-Osorio said earlier this year that he would be leaving the bank once a successor was found.
Bester's achievements during his short tenure include a transformation of the bank's IT systems and its separation from The Co-Op Group as well as improvements in customer satisfaction levels, the bank said.
"My ambition was to complete the major transformation phase of the turnaround and for our franchise to show resilience," Bester said in the statement.
"At this point, I believe the bank is on the right path and it is time for a new CEO to continue the journey to be the digital ethical bank."
(Reporting By Sinead Cruise, editing by Pamela Barbaglia)
