It's not essential, however; what I'd highly recommend is running a sponge filter on top of your existing filter, as if you have a power outage or filter break, the sponge will act as a back-up. This is even better with a battery operated.
I use water pumps, hardly ever air pumps.
If I feel the need for extra agitation, I put a venturi tube on the influent of the water pump to the tank. Venturis have an aperture that uses the power of the flow of water to suck air (or ozone, and other gases (Bernoulli's theorum) into the flow, and push the bubbles out into the tank. Many power heads have venturis built right in.
Another factor her would be what kind of filter are you using. If it is a HOB you probably will be fine without the air pump. But a canister putting your return back in the tank with little or no surface dsturbance the air pump would be a good idea.
I personally like to see more airation on most tanks. How ever some fish do need calmer water. What kind of fish are in your tank?
Are you just talking for aeration into the tank? Not necessary if you have enough surface agitation to get some O2 into the water. I personally use them on all my tanks just because I don't use powerheads in every tank