All solutions to Thomas’ family of Thue equations over imaginary quadratic number fields

We completely solve the family of relative Thue equations x3−(t−1)x2y−(t+2)xy2−y3=μ, where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. This is achieved using the hypergeometric method for t≥53 and Baker’s method combined with a computer search using continued fractions for the remaining values of t.