We already have a conjugation for the main verb (to see - "I saw") so the rest of the verbs don't conjugate for time. You can use "I saw him swim" or "I saw him swimming". It's not possible to use "swam".

I saw him swimming in the pool just now.

It really depends on what you MEAN. But they all generally mean the same thing. To extend it a little bit more: If you mean to say you saw him swimming in the past (leading into the present): I saw that he swam in the pond just now. If you mean to say you saw him swimming in the pond (an not in something else): I saw that he swims in the pond just now.

'swim' or 'swimming' is fine and more usual but you could say something like 'did you know he just swam in that pool?' or 'he just swam in that'.