As David Speyer has already suggested, we have $Q_f^\mathbb{C}\simeq Q_f\otimes\mathbb{C}$, so that answers the first question (and the second question).

For the third question, consider $f = (x^2+y^2)^2$. The singularity at $(x,y)=0$ is clearly isolated, but it is not algebraically isolated, since $\mu_\mathbb{R} = \infty$. Of course, over the complex numbers, $(z^2+w^2)^2$ has a non-isolated singularity at $(z,w)= (0,0)$.

The various confusions will be cleared up by noting that the ring of germs of analytic functions in two variables is a UFD, and, the singularity will be algebraically isolated as long as $f$ does not have any repeated irreducible factors. 'Isolated' is weaker over the reals simply because all the factors of $f$ might have no real zeros other than at the origin.

As David Speyer has already suggested, we have $Q_f^\mathbb{C}\simeq Q_f\otimes\mathbb{C}$, so that answers the first question (and the second question).

For the third question, consider $f = (x^2+y^2)^2$. The singularity at $(x,y)=0$ is clearly isolated, but it is not algebraically isolated, since $\mu_\mathbb{R} = \infty$. Of course, over the complex numbers, $(z^2+w^2)^2$ has a non-isolated singularity at $(z,w)= (0,0)$; the two lines of singularities $z = \pm i\,w$ intersect at $(0,0)$.

The various confusions will be cleared up by noting that the ring of germs of analytic functions in two variables is a UFD, and, the singularity will be algebraically isolated as long as $f$ does not have any repeated irreducible factors. 'Isolated' is weaker over the reals simply because all the factors of $f$ might have no real zeros other than at the origin.