Given an algorithm, is it possible to find all other equivalent algorithms for the same computable function in the same model
Interpreting your problem as finding a computable $f$ such that $f\big(\langle M\rangle\big)=\langle M'\rangle$ with the property that $M'$ enumerates all programs equivalent to $M$, i.e. all $M''$ with $L(M'')=L(M)$, the answer is no (such $f$ does not exist).
One way to show this is to observe that such $f$ would place the language $\left\{\big(\langle M_1\rangle, \langle M_2\rangle\big) \big| \; L(M_1)=L(M_2)\right\}$ in $RE$, while it is known that this language is $\Pi_2$ complete and thus not in $RE$. You could formulate this relative to any admissible numbering, and not necessarily in the language of Turing machines.