With a little more than a month to go until Elon Musk is supposed to abide by an agreement with the Securities and Exchange Commission and step down as Tesla’s chairman, speculation is growing over how Tesla might seek to fill the post in what has been one of the most tumultuous periods in the electric-car maker’s history.

Musk agreed to give up Tesla’s chairmanship for at least three years as part of a deal to settle fraud charges with the SEC related to statements he made in August about having secured funding to take Tesla private for $420 a share. The settlement, announced on Sept. 29, allowed Musk to remain as Tesla’s chief executive, but has Musk, and Tesla, each paying fines of $20 million to the SEC.

U.S. District Judge Allison Nathan still needs to give her approval to the settlement, and has given Musk and the SEC until Oct. 11 to submit a 10-page, double-spaced joint letter explaining why she should sign off on the agreement.

Musk was given 45 days from the day the SEC settlement was announced to step down as Tesla’s chairman. But the fact that Musk will still oversee the day-to-day running of Tesla has some analysts and Tesla watchers saying the company needs to be thorough when choosing a chairman who will, in theory, be Musk’s boss, but will have to work with all the drama that Musk can create.

“Tesla must name a new chairman that can be effective with the train wreck X factor potential of Musk,” said Eric Schiffer, CEO of private investment firm the Patriarch Organization. “They shouldn’t move too fast because it will take time to find, and then especially to persuade the right person that he or she isn’t going to be entering the realm of the impossible.”

There is also debate over what kind of background and experience Tesla’s next board chairman should have. This is seen as critical because of how Tesla straddles the line between being an automaker and a technology company that just happens to make electric cars as its main product.

“They desperately need automotive experience on that board,” said Rob Enderle, president of tech research firm the Enderle Group. “Much of the reason Tesla has been unable to get their (product) quality and quantity up to expected levels is the firm lacks high level automotive experience.”

Enderle said a potential chairman option can be seen in former Ford CEO Alan Mulally, who led Ford from 2006 until his retirement in 2014. Under Mulally’s leadership, Ford was the only major U.S. automaker to not take a bailout from the federal government during the Great Recession of 2007-2009.

“He handled a lot of the controversy at Ford well,” Enderle said. “He is retired but still young enough to serve, and he has the strength to say ‘B.S.’, even to Musk.”

Tim Bajarin, director of tech consultancy Creative Strategies, agreed that someone “having leadership skills and experience in strong corporate stewardship” should be at the top of Tesla’s search list. “Given some of the operational problems of the past, I would suggest that the new chairman really understand the (corporate) supply chain.”

But when it comes to speculating about who might be the right fit for Tesla, Bajarin said, “Besides (Apple CEO) Tim Cook, who already has a job, I have no ideas at the moment.”

The ability to relate to Musk may end up being the deciding factor in choosing Tesla’s next chairman, no matter how long the company takes before making its announcement public.

“Finding the right person is more important than anything in the long run and this, unfortunately, may take time,” said Clement Thibault, senior analyst with Investing.com. “For that chairman to bring more accountability to Musk, and the company, they simply must have a good relation. Of course, someone with the right experience can only help, but Tesla needs to minimize the fireworks. Inevitable clashes will need to be managed.”

Tesla has not returned a request for comment about whether it has begun its search for a new chairman.