Taking zinc on its own for a long stretch can quietly pull your copper down, because the two minerals compete in the gut. For short courses, say a week or two of zinc at the first sign of a cold, this does not matter. For months of daily zinc, the copper question is worth understanding.
Why zinc lowers copper
High zinc intake prompts cells lining the gut to make more of a protein that grabs onto copper and holds it there. Instead of being absorbed, that bound copper is shed as the gut lining turns over, and lost. Over months, a steady high zinc intake with no copper alongside it can tip someone into copper deficiency, which shows up as things like unexplained anaemia or nerve symptoms that look like something else entirely.
The ratio people quote
The figure you will see is somewhere around 8 to 15 parts zinc to 1 part copper, and it is a reasonable rule of thumb for long-term supplementing: if you take, say, 25mg of zinc daily for months, a small amount of copper (a couple of milligrams) alongside it keeps the balance. Many good zinc supplements include copper for exactly this reason, so it is worth checking the label before adding a separate copper tablet.
The important caveat
This runs the other way too. If you already happen to be low in copper, loading up on zinc is the wrong move, because it deepens the problem. So the honest advice is not "always add copper" but "match the two over the long term, and do not take high-dose zinc indefinitely without a reason". For a short cold-season course, zinc on its own is fine.
Zinc also competes with a few other minerals for absorption, which is part of the wider question of which supplements to space apart. To check zinc against any medication you take, the free checker will show you the result and the reasoning.