  This paper is continuation of the paper "Primitive roots in quadratic field".
We consider an analogue of Artin's primitive root conjecture for algebraic
numbers which is not a unit in real quadratic fields. Given such an algebraic
number, for a rational prime $p$ which is inert in the field the maximal order
of the unit modulo $p$ is $p^2-1$. An extension of Artin's conjecture is that
there are infinitely many such inert primes for which this order is maximal. we
show that for any choice of 85 algebraic numbers satisfying a certain simple
restriction, there is at least one of the algebraic numbers which satisfies the
above version of Artin's conjecture.
