Does there exist equivalent hash function? Is it possible to find equivalent hash function? Under equivalent, i assume same input produce same output, but they have different initial integer values and/or different set of bitwise operators. Or in other words, does there exist another algorithm of hash function, but having same input->output mapping. If exist, how to find or construct such algorithm, if not exist, do we have some proof of it? For example, we have different search algorithms, they have same input->output mapping, but different properties, for example complexity. Does this analogy work for hash function?
If a hash function can be described using boolean algebra, you can transform it into an equivalent function using distributivity, commutativity, associativity, De Morgan laws, absorption, etc... Two functions will be equivalent if you can find a serie of transformations that will lead to one another while preserving the equivalence. 

See:
https://en.wikipedia.org/wiki/Logical_equivalence

If you need a more complex function with flow control, whether two Turing machines are equivalent is undecidable in the general case. You might construct two functions with the same input/output mapping without being able to prove it except by testing all possible inputs, which hashing functions are designed to prevent. However, yes, you can always construct an equivalent function by yourself, by replacing subparts which are provably equivalent ($y+y+y$ becomes $y*3$) or reorder instructions when the order is commutative.