When R is the reals and x = i, the result is the complex numbers, a quadratic commutative extension of the reals inside the quaternions. If x is set to j or k, the same thing happens. Even x = (i+j+k)/sqrt(3) generates a ring that is isomorphic to the complex numbers.
In general, we always get a commutative quadratic extension. Write x as a+bi+cj+dk and square it, giving:
a2-b2-c2-d2 + 2abi + 2acj + 2adk
Subtract 2ax from the above and find an element in R. Thus x, no matter what it is, satisfies a quadratic polynomial, and R[x] is a commutative quadratic extension. If a, the constant term of x, is 0, then x is the square root of something in R.
Let p(x) be the quadratic polynomial associated with the quaternion x. If p is irreducible and R is a field, then R[x] is also a field. For instance, adjoining k to the reals gives the complex numbers.