You generally assume for big O purposes, when analyzing sorts for example, that comparisons of elements are constant time, and that swapping elements is constant time.
On an n-bit computer, when dealing with m-bit elements, those assumptions are broadly sound. Comparing two ints doesn’t depend on the ints. Reading or writing the int from memory doesn’t take more or less time either.
But the same algorithm, while it has the same big O relationship to the size of a collection, might have a different constant factor on different CPUs and for different values of m. And you might find that some algorithms that constant factor’s relationship to m has its own big O.
One common place this can bite you is when you try to apply big O analysis that has ‘constant time comparison’ assumption built in to, say, sorting arbitrarily long strings, or inserting into a hash table using arbitrary string keys. Comparing strings or calculating hashes of them is not constant time.