Azis denied that his departure is a result of HHCL's poor performance over the past 12 months, which has seen multiple redundancies and the loss of bedrock clients including Tango and Egg.
He said: "HHCL has had a tough time and will get through it, but I've come up with this idea now. I don't want to work in another ad agency, I want to help create a new way of marketing."
Fuller and Azis plan to create content and act as intermediaries between clients, production companies and media owners. The pair are seeking two other partners, one with media expertise and one from a creative background, to generate content ideas.
Fuller said: "We've found that people are biting around the edges (of content creation), but no-one has taken the bull by the horns. There's nobody specialising like this."
Azis added: "The convergence of the media, entertainment and communication business is throwing up so many creative opportunities. I want to be part of making that happen on a global basis."
Azis became the chief executive at HHCL in 1999. Since then he has been instrumental in moving the business in new directions, launching the digital offshoot Heresy and a direct marketing operation.
He has focused on the agency's tie-up with an international network for the past year, since the arrival of the chairman, Simon Burridge.
Jane Fuller Associates will continue as a production company. Fuller said: "For JFA it's business as usual, but the new venture may complement our existing business."
Fuller used to work in TV at HHCL where she and Azis both worked on the Tango campaign.
Azis' responsibilities will be shared by Burridge and the managing director Nick Howarth.
Burridge said: "We are very sad to see him leave. He has made a huge contribution over the years and we wish him the best in the future."