Well first off what exactly does this serum do you say it's both immortality and a super soldier serum but what does that actually mean?
1) Live forever.
2) Stay young forever.
3) Stay young forever +get maximum stat's.
4) Stay young forever and can NOT be killed, at all, in any way, PERIOD.
5) Stay young forever, get maximum stat's, can not be killed, become a mage.
And so on its hard to offer suggestions for side effects without knowing what the actual/intended effects will be. For example not being able to die takes on different meanings depending on whether the person involved is/is not able to regenerate severed limbs/nerve damage, is/is not aging, is/is not immune to diseases. More concretely compare immortal being who can't catch any diseases vs one who's spent the last 30 years suffering from one their body can't combat but they just can't die from even though it would have killed anyone not immortal (fever, hallucinations, vomiting, diarrhoea, boils, etc).