In answer to the question Demystifying complex numbers, Charles Matthews suggests "finding the points at twice the distance from (-1, 0) that they are from (1, 0)." as a motivation for complex numbers.

Suppose you want to find these points in hyperbolic geometry instead of euclidean geometry. If this can be done with vectors or complex numbers in R^2, then I reckon it could be done with gyrovectors or gyro-complex numbers in the hyperbolic plane, but if you don't use gyro-algebra then how would you find (or describe) the points at twice the distance from (-1, 0) that they are from (1, 0)?

(Defining coordinates in hyperbolic geometry can be done with gyro-algebra, but without it just assume the origin is an arbitrary point, and that (-1,0) is a point of distance 1 from the origin and (1,0) is a point of distance 1 from the origin in the opposite direction.)