The two colors on either side of the complement are called tertiary colors. A tertiary color is produced with a mixture of a primary colorand a secondary color adjacent to it on the color wheel. A tertiary color can also be produced with two secondary colors.
Combinations of two tertiary colors are called quaternary colors. Tertiary (or quaternary) colors can add sophistication to design.
A couple of gemstone combos:
Amethyst with new Burma jade and yellow jade
Lapis with tiger eye and matte carnelian
Apple jasper with jade and blue crazy lace agate
Crystal mixes:
Sapphire, Light Siam and light topaz
Fuschia, peridot and mint alabaster
Jonquil, tanzanite and rose
Looking for more examples of split-complementary color mixes? Try Pinterest. I searched "split-complementary color schemes" and found pages of inspiration.
Some are beads, some are fabric and others are paintings or interior decoration. But they can all be translated into a bead design.
In addition to Pinterest, there is a bounty of sites that discuss color mixing – most are focusing on painting and interior decorating, but they can easily be translated to jewelry-making elements.
Another three-color mix to experiment with is a triadic color scheme. A triadic color scheme uses three colors equally spaced around the color wheel. Triadic color schemes tend to be vibrant, even when using paler tints of the colors in question.
This option gives offers strong contrast paired with harmonious elements. The triadic mix doesn’t have quite the pop of a complementary scheme, but it is relatively easy to create balance and harmony.
The easy examples of color triads are: Red, yellow and blue (primary), and orange, green and violet (secondary). Other triads can be less obvious. One foolproof way to find more subtle triads is to place an even-sided triangle on your color wheel.
The triangle should be slightly smaller than the wheel. Obviously, the color wheel you use would have to go beyond primary and secondary colors. You can find one by Googling "color wheel" images. Or you can find a wide selection on Amazon or other websites.