After the worst December since 1931 and the worst year since the financial crisis, stocks enter 2019 tentatively but still susceptible to the volatility that resulted in historic intraday swings in late December. The S&P 500, trading at about 2,495, is down 6.6 percent for the year, and down 9.6 percent in December alone.

In the final quarter of 2018, the market fell on concerns that economic growth and profit growth are slowing down, and Fed rate hikes and trade wars would only accelerate the deceleration. But most Wall Street strategists in a CNBC survey maintain a rosy view for the bull market, and see it extending its run for another year, for an average year-end target of 3,000 — a 20 percent gain.

"Based on fundamentals, I don't think the pullback we had in this market was ever justified. Markets will do what they'll do. I think you have significant upside here. Therefore, we would think that the bottom has been put in this market," said Jonathan Golub, chief U.S. equities strategist at Credit Suisse. One positive is that the forward price to earnings ratio has fallen below 15 percent from 18.4 at the start of 2018. Golub's S&P 500 target is 2,925 for2019.

But while recent up days have been encouraging, some strategists say the market could stay volatile and swing in both directions as it works to form a bottom around the lows of last week.

Ed Keon, chief investment strategist and fund manager at QMA, is clearly feeling better about the market but says it continues to have big risks, and he expects earnings growth to be flat in 2019, compared to Wall Street expectations of about 7 to 8 percent growth.

Fed outlook

The stock market has been fearful of the Federal Reserve's rate hiking agenda, but in the futures market, there is not a single hike in the federal funds rate priced in for next year. For 2020, the market is figuring there's a higher chance for a rate cut than a hike.

"In 2018, we were facing a Fed that was tightening monetary conditions. Next year, the Fed will probably be finished raising rates. What the market is struggling with is it's a Fed that says one thing. Then if you look at inflation, which has gotten weaker, the market is basically saying the Fed is kind of done now," said Golub. "The Fed is signalling something that is out of sync with the market.That's what the market is struggling with...The market believes the Fed is going to be done in 2019. That s a huge positive over 2018."

Economists expect inflation comparisons should weaken, given the near 40 percent decline in oil prices in the fourth quarter.

Trade outlook

As for trade, President Trump tweeted some encouraging developments over the weekend, though it's unclear how much progress is being made. There is a March 1 deadline on a trade deal, or Trump has said he would resume putting tariffs on China.

if we were to reach a lasting fair and equitable peace on the trade war front that would be good news," said Keon, noting it would be a positive catalyst for the market.

"It's such a wild card," said Golub.

But there is skepticism around a deal being made in time to head off further tariffs. Goldman Sachs economists, for instance, lowered their growth forecasts to 2 percent for the first half of the year, and noted that more tariffs are likely.