The exit of Tesla Inc.’s TSLA -10.34% accounting chief on Friday, after only a month on the job, places a spotlight on the high turnover of executives at the electric car maker. It also highlights the challenges the company could face attracting and retaining talent amid increased regulatory scrutiny and recent controversial actions of its founder.

More than 50 executives have departed the company during the past 24 months. Chief Accounting Officer Dave Morton is among the latest.

“The level of public attention placed on the company, as well as the pace within the company, have exceeded my expectations,” Mr. Morton said in a securities filing. “As a result, this caused me to reconsider my future.”

He didn’t respond to a request for comment. A Tesla spokesman declined to comment beyond the company’s regulatory filing. Tesla’s share price approached its 2018 low on Friday.

Mr. Morton’s tenure at Tesla coincided with an unusual bout of public scrutiny. He joined the company on Aug. 6, one day before Chief Executive Elon Musk used social media to float the prospect of taking Tesla private. The message on Twitter triggered gyrations in Tesla’s stock and sparked a U.S. Securities and Exchange Commission investigation.

The SEC last year began investigating whether Tesla misled investors about its Model 3 car production problems, The Wall Street Journal reported in August. That probe was launched before regulators started looking into Mr. Musk’s tweets about taking the company private.

Meanwhile, Mr. Musk’s behavior continues to draw concern: on Thursday, he appeared to smoke marijuana during a live interview in which he spoke on wide-ranging topics for more than 2 ½ hours.

These issues would likely cause quality candidates to think twice if approached to fill executive roles at the company, executive recruiters said.

“In a market like this, where there’s very high demand and people don’t need to leave, if you have anything that causes concern or sensitivity at the top, the candidate pool goes, ‘Whoa, I don’t need that,’” said Peter Crist, chairman of executive recruiter Crist | Kolder Associates.

Tesla, in its filing, said accounting functions and personnel would be overseen by its Chief Financial Officer Deepak Ahuja and its corporate controller, “as had been the cases prior to and during Dave’s transition to Tesla.”

Mr. Musk has said that executive turnover at the company is in line with that of other large companies and has announced plans for a reorganization aimed at flattening the layers of managers. On Friday afternoon, he announced a slew of promotions in an email to employees. Tesla has also hired new talent in recent months, naming eight new executives, including a new CFO for China, James Zhou, who had joined from Ingersoll-Rand PLC.

A chief accounting officer is often in charge of certifying financial statements that are submitted to regulators, which means ensuring the books are above reproach. Tesla’s Mr. Morton brief tenure suggests he had not fully taken on those responsibilities.

The cautious, reputation-conscious nature of finance professionals could complicate the recruitment of Mr. Morton’s replacement. Accountants often follow strict professional ethics codes and may worry about working for a company that is being investigated by the SEC — regardless of the outcome.

“The finance community, by nature, they’re risk averse,” Mr. Crist said. “If you’re a CFO and somebody is coming after you with an opportunity, you’re going to do a pretty deep risk assessment of the situation, because you don’t want to have happen what happened today, it’s just easier to say ‘no thanks.’”

Mr. Morton’s predecessor resigned in March after 18 months. Former CFO Jason Wheeler left Tesla in early 2017 after less than two years in the role. He was succeeded by Mr. Ahuja, a company veteran who returned from retirement.

Such moves aren’t without precedent, and going without leadership in key finance roles doesn’t necessarily spell a crisis for a company.

Uber Technologies Inc. last month filled its CFO post, which had been vacant since 2015. The ride-hailing services company in early 2016 said it wasn’t looking for a CFO. But the search was launched in earnest shortly after Chief Executive Dara Khosrowshahicame on board in 2017, replacing founder Travis Kalanick who left under a cloud of scandals.

Mr. Khosrowshahi’s arrival at the helm of Uber likely helped drive a positive outcome for the company, Mr. Crist said. “If you’re talking about building a finance organization, you’re always talking about who’s at the top, who’s going to bring the talent in,” he said.

Still, Mr. Khosrowshahi faced his own setback. He had hoped to reach an agreement with VMware Inc. CFO Zane Rowe for the long-vacant job, but Mr. Rowe indicated he would turn it down, The Wall Street Journal reported in May. A spokesman for Uber declined to comment.

Candidates considering a role with a company like Tesla or Uber can ask for “danger money,” said Mark Freebairn, head of the financial management practice at recruitment firm Odgers Berndtson.

“These businesses will have to be flexible to attract the level of talent they want,” he said. “If they want someone really badly, they will have to offer a package that compensates for the potential risk to their reputation.”

Finance executives prize their reputation above anything else, he said.

“Elon could blow up that business now, in five minutes, today, in a year—and you as accounting or finance chief are tied to that,” he said.

Mr. Freebairn said he struggled to place former Tesco PLC finance staff member after the company’s 2014 accounting scandal. “Every time I put forward someone from Tesco, I got questions about a potential involvement in the accounting issues,” Mr. Freebairn said. The U.K. grocer in September 2014 announced it had overstated profit by £263 million ($340.5 million).

Write to Tatyana Shumsky at tatyana.shumsky@wsj.com and Nina Trentmann at Nina.Trentmann@wsj.com