There is a similarity between resolving complex numbers and radicals into standard form.
First similarity is to reduce the radical to its simplest form. Take √y where y can be expressed as n²x, that is the product of a number (which is not a perfect square) and one that is. √y is reduced to n√x. This is analogous to √-y which reduces to i(n√x).
The second similarity is the use of conjugates. The conjugate of the complex a+ib is a-ib. The conjugate of a+√b is a-√b. To radicalise (simplify) (c+√d)/(a+√b), multiply top and bottom by the conjugate of the denominator, a-√b: (c+√d)(a-√b)/(a²-b)=(ac-√(bd))/(a²-b). This is now in standard form for the expression of an irrational number.