The European Central Bank’s big stimulus plan has thrust the dollar’s advance into overdrive, for better or worse.

The Wall Street Journal dollar index, which tracks the value of the dollar versus 16 other currencies, is up 28% since its postcrisis low in July 2011. This is thanks to an improving American economy, expectations for higher U.S. rates and ultralow ones in Europe and Japan and a few minicrises in emerging markets such as Russia and Turkey.

The biggest gains may yet be to come. With the ECB preparing to pump €60 billion ($ 68.21 billion) a month into markets “at least” until September next year, investors and analysts are contemplating a broad, cyclical upturn for the dollar like those seen in the 1980s and 1990s.

A replay would bolster U.S. stocks but could come alongside increasing volatility that ultimately could threaten global financial stability.

A rising dollar would keep U.S. inflation down, providing a boon for consumers, while making U.S. exporters’ products much more expensive overseas. Dollar strength will eat into returns in foreign markets for U.S. investors and could make it costly to hedge against foreign-exchange losses.

In the short term, said Steven Wieting, global chief investment strategist for Citi Private Bank, ECB monetary easing is good news for riskier assets such as stocks. But he notes that many overseas gains are likely to be offset by local-currency weakness, as they largely have been in Japan in recent years.

A 6% rally in eurozone stocks since the ECB decision has already been offset by the euro’s decline. That’s a reminder that “U.S. dollar-based investors can only accrue these strong local market returns on a currency-hedged basis,” he said. Citi Private Bank has $ 310 million under management.

A key question, said Kit Juckes of Société Générale, is what sort of investor sentiment is behind the gains.

He wants to see a “good dollar” rally, in which an improving U.S. economy would drive the dollar higher while bolstering investor sentiment globally.

More concerning would be “bad dollar” rally, in which the dollar strengthened because of growing concerns about the health of economies and markets overseas. That would result in “much bigger moves,” as were seen in the 1990s, “when global capital asset flows were utterly disrupted and American investors lost faith in foreign assets,” said Mr. Juckes.

Such a global meltdown in confidence isn’t Sociéte Générale’s central expectation. Mr. Juckes still has a formal year-end forecast of 1.10 for the euro but, like many analysts, now believes a drop to parity is likely.

In the worst case, he said, the dollar gains a further 30% on a trade-weighted basis. Such a rally could end in a replay of the 1998 crisis that featured a wholesale exit from equities and other risk assets.

For now, large dollar gains seem almost inevitable. Nearly all major central banks outside the Federal Reserve are biased toward easier monetary policy, in part because they want their currencies to be unappealing to investors fleeing euros.

In recent weeks, the central banks of Switzerland and Denmark pushed rates deeper into negative territory, the People’s Bank of China weakened the yuan, the Bank of Canada cut rates for the first time in 4½ years and the Bank of England revealed that two members of its policy committee who had previously voted for higher rates had changed their minds.

By contrast, the Fed is on track to raise rates later this year, in keeping with the idea that the U.S. economy is in far better shape than deflation-gripped Europe and Japan.

Deutsche Bank strategist Alan Ruskin said the U.S. currency’s gains have so far been mostly driven by reactions to news from other countries, highlighted by the dollar’s giant spike against the euro in reaction to a mostly anticipated policy shift by the ECB.

But “if we do get higher U.S. rates,” he said, “it will bring a cyclical kick and a more dollar-centric leg to what has already been a strong dollar story.”