Put the clam juice in a medium microwave-safe bowl and microwave until very hot, about 5 minutes.
Meanwhile, scatter the potatoes, carrots, celery, onion and garlic in a 6-quart slow cooker. Tie the dill, parsley, thyme, bay leaves and lemon zest together with twine and add to the slow cooker. Pour in the hot clam juice; cover and cook on high, 3 1/2 hours.
Season the salmon with 1 1/2 teaspoons salt and 1/2 teaspoon pepper. Stir the heavy cream into the vegetables in the slow cooker, then partially submerge the salmon in the liquid. Cover and cook 30 more minutes.
Discard the herb bundle; season the chowder with salt and pepper. Serve topped with chives and dill.