We're getting ever so close to a new bull market. Another few days of gains for the S&P 500 could officially put the major index back in bull territory. But as the old saying goes, "Close only counts in horseshoes and hand grenades." There's no guarantee that the S&P will move the 4% or so higher needed. 

The good news for investors is that there are great stocks that aren't dependent on the S&P 500's gyrations. Here are three stocks that could soar whether there's a new bull market or not.

1. Axsome Therapeutics

Axsome Therapeutics (AXSM -0.39%) stock more than doubled last year while the S&P 500 plunged. So far in 2023, though, it's a different story. Axsome's share price has fallen as the S&P rose. But don't read too much into this seemingly inverse relationship.

The reality is that Axsome's performance this year won't hinge in the least on what the overall stock market does. What will matter for Axsome is how its commercial launch of Auvelity in treating major depressive disorder (MDD) goes and how much progress its pipeline programs make.

Investors will get an initial answer about Auvelity when Axsome reports its 2022 fourth-quarter update on Feb. 27. I expect good news. Auvelity is the first new type of oral therapy approved for MDD in more than six decades. It's also the first and only rapid-acting oral antidepressant. My hunch is that the drug will enjoy a strong launch. 

I also look for solid pipeline progress this year. Axsome plans to refile for U.S. approval of AXS-07 in treating migraine and submit for U.S. approval of AXS-14 in treating fibromyalgia. It's scheduled to announce results from a late-stage study of AXS-12 in treating narcolepsy as well. 

A new bull market could begin soon or another pullback could be on the way. This biotech stock, though, should resume its upward momentum with a positive Q4 update and no pipeline setbacks.

2. Occidental Petroleum

You can probably guess what factor is most important for Occidental Petroleum (OXY -0.29%) stock's performance in 2023. If oil prices rise, so will the stock. 

Goldman Sachs thinks that oil prices will move above $100 per barrel this year. Sanctions seem likely to negatively impact Russian exports. China's shift from its Covid-zero policy should drive higher demand. The laws of supply and demand should lead, therefore, to higher oil prices. 

Occidental would be a key beneficiary if this scenario unfolds. Even though the stock is up more than 120% since the beginning of 2022, investors should still find its valuation attractive. Occidental's shares trade at less than 8.4 times expected earnings.

There's another factor that could work in Occidental's favor, too. Warren Buffett loaded up Berkshire Hathaway's portfolio with the stock throughout most of last year. Buffett is almost certainly continuing to buy Occidental in 2023. Berkshire won regulatory approval to acquire up to 50% of the company, but it currently owns only 21.4%.

3. Vertex Pharmaceuticals

Last, but certainly not least, is my favorite biotech company on the planet -- Vertex Pharmaceuticals (VRTX 0.32%). Last year, Vertex's shares jumped more than 30% as the overall stock market sank. I believe that the big biotech stock could crush the market again in 2023.

Vertex's monopoly in treating the underlying cause of cystic fibrosis (CF) isn't going away anytime soon. That means the company can count on a steady revenue stream and high profits. 

Even better for the stock's near-term fortunes, though, is that Vertex could soon have a new therapy outside of CF. The company and its partner, CRISPR Therapeutics, hope to win U.S., U.K., and European regulatory approvals in the near term for exa-cel in treating sickle cell disease and beta-thalassemia.

If this was an infomercial, it would be time for the exclamation, "But wait, there's more!" Vertex's pipeline also includes two other programs that could win regulatory approvals relatively soon -- non-opioid pain drug VX-548 and the vanzacaftor triple-combo CF therapy. 

The S&P 500 will do whatever it's going to do. Vertex could have its own bull market no matter what.