bugcat: I had an idea for a coordinate system that uses the symmetry of the board to make it easy to recognise common moves around the corners. Each 9x9 corner area is split into two directions, which radiate from the diagonal corner axis into the centre. Indexes 1-4 are corner areas, 5-8 are centre lines, and 9 is tengen. The board is notated as so:
4-x1 4-A1 4-B1 4-C1 4-D1 4-E1 4-F1 4-G1 4-H1 8-1 1-h1 1-g1 1-f1 1-e1 1-d1 1-c1 1-b1 1-a1 1-x1
4-a1 4-x2 4-B2 4-C2 4-D2 4-E2 4-F2 4-G2 4-H2 8-2 1-h2 1-g2 1-f2 1-e2 1-d2 1-c2 1-b2 1-x2 1-A1
4-b1 4-b2 4-x3 4-C3 4-D3 4-E3 4-F3 4-G3 4-H3 8-3 1-h3 1-g3 1-f3 1-e3 1-d3 1-c3 1-x3 1-B2 1-B1
4-c1 4-c2 4-c3 4-x4 4-D4 4-E4 4-F4 4-G4 4-H4 8-4 1-h4 1-g4 1-f4 1-e4 1-d4 1-x4 1-C3 1-C2 1-C1
4-d1 4-d2 4-d3 4-d4 4-x5 4-E5 4-F5 4-G5 4-H5 8-5 1-h5 1-g5 1-f5 1-e5 1-x5 1-D4 1-D3 1-D2 1-D1
4-e1 4-e2 4-e3 4-e4 4-e5 4-x6 4-F6 4-G6 4-H6 8-6 1-h6 1-g6 1-f6 1-x6 1-E5 1-E4 1-E3 1-E2 1-E1
4-f1 4-f2 4-f3 4-f4 4-f5 4-f6 4-x7 4-G7 4-H7 8-7 1-h7 1-g7 1-x7 1-F6 1-F5 1-F4 1-F3 1-F2 1-F1
4-g1 4-g2 4-g3 4-g4 4-g5 4-g6 4-g7 4-x8 4-H8 8-8 1-h8 1-x8 1-G7 1-G6 1-G5 1-G4 1-G3 1-G2 1-G1
4-h1 4-h2 4-h3 4-h4 4-h5 4-h6 4-h7 4-h8 4-x9 8-9 1-x9 1-H8 1-H7 1-H6 1-H5 1-H4 1-H3 1-H2 1-H1
7-1 7-2 7-3 7-4 7-5 7-6 7-7 7-8 7-9 9 5-9 5-8 5-7 5-6 5-5 5-4 5-3 5-2 5-1
3-H1 3-H2 3-H3 3-H4 3-H5 3-H6 3-H7 3-H8 3-x9 6-9 2-x9 2-h8 2-h7 2-h6 2-h5 2-h4 2-h3 2-h2 2-h1
3-G1 3-G2 3-G3 3-G4 3-G5 3-G6 3-G7 3-x8 3-h8 6-8 2-H8 2-x8 2-g7 2-g6 2-g5 2-g4 2-g3 2-g2 2-g1
3-F1 3-F2 3-F3 3-F4 3-F5 3-F6 3-x7 3-g7 3-h7 6-7 2-H7 2-G7 2-x7 2-f6 2-f5 2-f4 2-f3 2-f2 2-f1
3-E1 3-E2 3-E3 3-E4 3-E5 3-x6 3-f6 3-f7 3-h6 6-6 2-H6 2-G6 2-F6 2-x6 2-e5 2-e4 2-e3 2-e2 2-e1
3-D1 3-D2 3-D3 3-D4 3-x5 3-e5 3-e6 3-e7 3-h5 6-5 2-H5 2-G5 2-F5 2-E5 2-x5 2-d4 2-d3 2-d2 2-d1
3-C1 3-C2 3-C3 3-x4 3-d4 3-d5 3-d6 3-d7 3-h4 6-4 2-H4 2-G4 2-F4 2-E4 2-D4 2-x4 2-c3 2-c2 2-c1
3-B1 3-B2 3-x3 3-c3 3-c4 3-c5 3-c6 3-c7 3-h3 6-3 2-H3 2-G3 2-F3 2-E2 2-D3 2-C3 2-x3 2-b2 2-b1
3-A1 3-x2 3-b2 3-b3 3-b4 3-b5 3-b6 3-b7 3-h2 6-2 2-H2 2-G2 2-F2 2-E2 2-D2 2-C2 2-B2 2-x2 2-a1
3-x1 3-a1 3-a2 3-a3 3-a4 3-a5 3-a6 3-a7 3-h1 6-1 2-H1 2-G1 2-F1 2-E1 2-D1 2-C1 1-B1 2-A1 2-x1
Not only can we tell that x4 is always a 4-4 and that x3 is always a 3-3; it would be easy to recognise e3 as an approach to a 4-4. A person could even memorise the sequence x4 - e3 - g3 - x3 as the start of a pincer joseki, then that sequence would look the same anywhere on the board.
Though I'll be honest, this would probably be an absolute pig to work with after the opening.