Jeff Bezos congratulates President-elect Trump, after offering to shoot him into space

Jeff Bezos
Gary Cameron | Reuters

After learning that Washington Post reporters were looking into his past, Donald Trump railed against Amazon chief Jeff Bezos in a tweetstorm. Bezos replied with #sendDonaldtospace. Trump warned, "If I become president, boy do they have problems."

Well, Trump will indeed become president — so how worried should Amazon be?

Investors are certainly nervous. Amazon shares fell as much as 6.5 percent since Trump's victory, while the broader markets rallied to record highs.

Trump suggested that Amazon had an antitrust issue, given the company's e-commerce dominance. He also suggested that Bezos' ownership of the Post was helping Amazon keep its taxes low. Bezos struck back in October, saying the then-Republican nominee's behavior was "eroding our democracy around the edges."

Bezos is now striking very different tone. On Thursday, he tweeted, "Congratulations to @realDonaldTrump. I for one give him my most open mind and wish him great success in his service to the country."

It's not clear what actions Trump could take to prove that Amazon is what he called "a big tax shelter" based on its relationship to The Washington Post — especially since Bezos personally owns the publication, not Amazon. But the IRS could increase its scrutiny of Amazon.

Trump's accusation of antitrust problems would also be difficult to prove as Amazon's business revolves around offering low prices. Antitrust laws aim to protect consumers from unfair price hikes where a company has a monopoly.

Piper Jaffray analyst Gene Munster said Amazon does not have to fear a Trump presidency, as President Trump will be very different from candidate Trump, and that "it was all probably a lot of hot campaigning air that Trump was letting off." He also noted that Amazon has more foreign taxes on its products than nearly any other American company.

And even if a Trump administration goes after Amazon on tax issues, Munster says there's nothing to find. "The reality is that Amazon doesn't do anything that's wrong."

But in a broader sense, there's the issue of what free trade will look like under Trump and how that could affect Amazon, which has been growing its footprint outside the U.S. In its latest annual letter, Amazon identified trade protection measures as a risk to its operations. That risk could grow if policies change under a more protectionist administration.