The 7th-level egoist power fission [psychometabolism] (Expanded Psionics Handbook 108) renders neither the egoist nor the duplicate immune to instantaneous effects, including, for example, most form hit point damage, and certainly not in a way that enables the power fission to be used somehow has an ersatz version of the 9th-level psion power timeless body [psychoportation] (XPH 138).
When the description of the 7th-level egoist power fission says says, "All powers affecting a fissioned creature, either the original or the duplicate, end when the fission ends," the description means that when the fission power's effect ceases—either because the fission effect's 1 round/level duration expires, the egoist dismisses the power, or the the power is somehow otherwise dispeled—, all psionic powers (and, via transparency, all spells) affecting the egoist likewise—suddenly and completely—also cease. (I suspect this ill-conceived sentence—absent from the description of the power fission in the earlier and largely obsolete Psionics Handbook—was added to the power's description to prevent the egoist from abusing fission by buffing half of himself at twice the speed then merging into a superbuff psion. Thank heavens the authors thought to prevent those shenanigans, huh?)
That means, for example, the 9th-level psion power apopsi [telepathy] (XPH 76) should continue to affect the egoist as the apopsi power's duration is instantaneous: the effect of apopsi can't normally cease. Similarly, for example, the 9th-level psion power microcosm [telepathy] (XPH 118) also has an instantaneous duration, so that power's effect, too, does not cease at the conclusion of the fission effect. But what happens to the egoist when he does rejoin with his duplicate if either he or his duplicate has been affected by an instantaneous yet not death-causing effect—like petrification or the reduction in height from the rod of wonder (DMG 237) (12,000 gp; 4 lbs.)—is a mystery the DM must solve. Neither FAQ nor errata even mentions the fission power.
I suspect most DMs will, if it's possible, average in the merged form instantaneous effects that were affecting the egoist and the duplicate separately (maybe slightly in the merged form's favor if the duplicate's affected by the effect as the duplicate is like the egoist except minus two levels). For purely binary instantaneous effects, this DM would likely give the egoist upon merging with the duplicate another saving throw against the instantaneous effect suffered by either duplicate or egoist (perhaps with a −2 penalty if it were the egoist affected by the instantaneous effect and with a +2 bonus if the duplicate were instead affected and perhaps even if the original effect didn't allow a saving throw), with success causing the merged form not to suffer the instantaneous effect and failure causing the merged form to suffer the instantaneous effect. This is easier than, for example, determining which parts of the merged form are now petrified or reduced in size.
Finally, there is a seeming contradiction in the fission power's description. Initially, the description says, "At the time of rejoining, you take half of the damage your duplicate has taken since this power was manifested." Then, later, the description says that when the power ends that "[a]ll damage, including hit point damage, ability damage, ability drain, and ability burn damage, is added together." I suspect that upon rejoining the egoist's merged form suffers half of the hit point damage, ability damage, ability drain, and ability burn damage that the duplicate suffered plus all of the hit point damage, ability damage, ability drain, and ability burn damage that the egoist suffered while the fission effect continued. There are better ways to phrase this concept, though.