Most accessories are interchangeable between hammock and brand but always be sure to read descriptions and reviews carefully lest you end up with gear you can't use.
Suspension: Most hammocks include carabiners or loops that are strength tested to be sufficient. In addition, you need a pair of tree straps, your suspension, to actually hold you off the ground. Suspension should be a minimum of 3/4 of an inch wide to avoid digging into the tree and causing damage. Leave no trace!
Warmth: The biggest insulation difference between hammock camping and tent camping is the need for an under quilt. When sleeping, there is nothing stopping cool air from chilling your backside. A sleeping pad slipped into a double layer hammock can help but your body will compress the pad and reduce the insulation. Your best bet is to use an under quilt. Most hammocks have clips that integrate with insulation or elastic cords you attach to your carabiners that hang your hammock.
Most experienced hammock campers also find that instead of a sleeping bag using a top quilt is more comfortable and doesn't restrict movement. Imagine trying to shimmy back into your mummy bag after a midnight bathroom break. Not fun.
Rain Fly: Similar to your tent, this protects you from rain, wind, and snow while camping. It can be attached to either your hammock suspension or have independent suspension and uses guylines to keep a tight pitch. A larger tarp will provide greater protection but may weigh more depending on the model.
Bug Net: Look for models that offer either 360 protection or have a thick enough denier in the fabric to avoid butt bites. Also, be sure that it has
"no-see-um" mesh (most do) to keep gnat sized biters out.
Gear Stash: Don't forget about your trusty pack and leave it out in the rain. The most convenient option is hammocks that have the gear stash integrated into the bug net but you can also find gear stashes that are either hung inside or underneath your hammock to keep your other gear clean.
These are the most common accessories, and arguably completely necessary for hammock camping. There are plenty of other fun accessories on the market though like hammock stands and lights to string along your ridgeline.