The consumer group said the Office of Fair Trading and Ofgem should refer the market to the Competition and Markets Authority for investigation into whether competition was working for consumers and how it could be increased to deliver trust, transparency and fair prices.
Which? has launched a campaign calling for "six fixes" to address trust in the market after figures from energy companies revealed they drew 5,579,665 complaints from customers.
Complaints ranged from issues with bills and metering to customer service, switching and payments.
Of 5,579,665 complaints in 2013, Npower received 1,383,650 - the most of all six firms. They were followed by EDF with 1,240,005, British Gas with 1,235,550 and E.ON with 929,230. SSE and Scottish Power received 482,582 and 308,648 respectively.
Which? said the latest complaints figures from the six companies - who handle 97% of the market - were "yet another indication that the energy market is not working for its customers".
It said it wanted to see an industry "that works for customers as well as shareholders, where prices are kept as low as possible and people can trust companies to give them a good deal".
Which? said the Government must control costs added to bills, while energy firms must ensure charges were cost-reflective and should not hold on to high levels of customers' cash when they were in credit.