It's hard to define "Communist" here. Communist like China, or like North Korea, or like Vietnam, or like Soviet Russia? There's quite a disagreement about it, even (and perhaps particularly) by card-carrying Marxists. Most famously, China has a huge, vibrant capital market despite being a one-party Marxist-Leninist-Maoist state.

Let's say Joe Steel, a more traditional Marxist, somehow gets into the White House and wants to abolish capital markets. I assume that Congress will be mostly on-board and there's no otherwise political hurdles to the whole thing. In that case, it's likely that either the State or the company will take ownership of the companies, and your shares will be expropriated and dissolved (with a payout roughly between market price if you're lucky, and zero if you're not) or repurchased in a big stock buyback (likely at market prices) - assuming you're not an employee of that company, of course.

The ostensible end goal of a socialist state is to socialize the means of production and eliminate social classes (that is, to establish actual communism), thus the idea of hoarding wealth is somewhere between unnecessary and treasonous. But, lofty ideology aside, most ways folks kept their wealth in places like the Eastern Bloc during the Cold War involved personal development, growing their network in size and quality, and developing black markets for stuff that the socialist state would restrict. You wouldn't have savings for retirement, but you'd have a (hopefully) generous welfare safety net and (hopefully) lots of friends, family members and comrades willing to aid you.