Can Turing machines simulate the unbounded minization operator applied to a partial function?
Unsafe $\mu$ is strictly stronger than partial TMs.

Indeed, define $f$ such that $f(x,0)=1$ if the $x$-th Turing machine halts on empty input, let $f(x,0)$ be undefined otherwise, and let $f(x,n)=1$ for all $n&gt;0$. Then $f$ is a computable partial function. Let $g:=\mu(f)$. Then $g(x)=0$ if the $x$-th Turing machine halts and $g(x)=1$ otherwise. Hence $\mu(f)$ is not computable.

[EDIT]

The closure of partial recursive functions under unsafe $\mu$ seems to be the arithmetical hierarchy. (In order to make that statement precise, we have to translate between partial functions and subsets of $\mathbb N$.)