Notation for a Kleene fixed point with a starting element
Suppose $Q$ is a join semilattice.  Define $g:Q \to Q$ by

$$g(x) = f(x) \sqcup q.$$

Then $\text{fix} \; g$ is equal to the least fixpoint of $q,f(q),f^2(q),\dots$.  So, at least for cpo's that form a join semilattice, you can indicate it pretty naturally using existing notation -- there's not a strong need to invent a new notation (though you could of course introduce one if you wanted).

If $Q$ is not a join semilattice, I think you can still define

$$g(x) = \sup \{ f(x), q \}$$

and then do the same thing.  The least fixpoint of $q,f(q),f^2(q),\dots$ will be $\text{fix} \; g$.