Mike, despite agapanthus producing such a wonderful root system, I'm afraid they cannot be propagated from root cuttings. Division is probably your best way of raising identical plants to the parent, and plants can be raised from seed, although may look different from their parent.
Root cuttings are not a common form of propagation, but can be used for propagating oriental poppies, Phlox paniculata, rhus, and other shrubs and perennials. Mint and other plants that spread by root runners can also be propagated in this way.