The stock market bounce following the deal avoiding the fiscal cliff will probably be brief, as financial markets face a rough two months ahead, experts told CNBC.

Congress begins its true test when it starts talks over increasing the government’s debt ceiling. And the stakes are even higher. Failure to raise the limit would mean government default.

“This will plague us at least until the end of the first quarter because we have got these discussions about raising the debt ceiling, which will lump all negotiations — expenditure, tax hikes — into one,” Andrew Economos of JPMorgan Asset Management told CNBC.

Video: Economist Predicts 'Unthinkable' for 2013

“So the first quarter is going to be choppy.”

Martin Lakos, division director of Macquarie Private Wealth in Sydney, Australia, agrees.

“I think the problem for markets going forward is the fact that we are really not addressing the debt ceiling issue until the 28 February deadline,” he told CNBC. “So unfortunately, we are going to get a couple of months of volatility.”

Although the settlement avoids the fiscal cliff by making Bush-era tax cuts for most workers permanent, analysts say it does not solve the country’s long-term budget problem, according to CNBC.

Markets fear a replay of the protracted political fight in 2011. Congress raised the debt ceiling at the last minute only after agreeing to future deep spending cuts.

Many experts are disappointed that the fiscal cliff settlement is not a “grand bargain” that tackles the country’s real budget issues.

With Republicans against tax increases and Democrats against cutting healthcare or Social Security, prospects for quick agreement appear dim.

“Commonsense indicates they should just meet in the middle on both tax and spending,” Shane Oliver, chief economist with AMP Capital, told CNBC. “But unfortunately the failure to do so means another contentious debate that will likely take us to the brink of a government shutdown and possible ratings downgrades in February.”

The budget deal does little to improve the economy, argues Peter Morici, a professor at the Robert H. Smith School of Business at the University of Maryland and former chief economist at the U.S. International Trade Commission.

Huge trade deficits with China and oil exporters are contributing to weak demand and slow job creation, Morici writes in his blog. Plus, more business regulations, rising healthcare costs and higher taxes will crimp economic growth.

“Most analysts see the unemployment rate inching up to 7.8 percent, while a few see it remaining steady,” he writes.

Video: Economist Predicts 'Unthinkable' for 2013