A simplem-primary complete ideal in a two-dimensional regular local ring (R,m) is associated to a unique prime divisor ofRand to a unique point infinitely near toR. It also has a unique complete ideal ofRwhich is adjacent to it from above. This latter uniqueness can be extended to ideals which are products of more than one distinct simple complete ideals in certain cases. It is also known that there are infinitely many complete ideals adjacent to any power of a simple complete ideal. In this paper we extend the uniqueness as well as the existence of infinitely many complete ideals adjacent to certain ideals which are products of more than one distinct simple complete ideals such that at least one of them has exponent greater than one in the unique factorization.