Some countries do ask for something of the kind already, giving a free passage for people who already have had COVID-19. That even if it been a mild case, which is no guarantee for getting infected again and so become a carrier.
In Norway they give a freebie if you go trough a simple swab test, such swab tests were demonstrated to be highly unreliable with less than 40% or less already during the first outbreak in Wuhan and proved once again when Korea got struck in the very early part of this when it were not even considered a pandemic.
The worst part that anything such only will boost the market for those who already sell forged documents, only use any search engine of your choice and you'll see that it's possible to get documents of any kind for a price.
The image I show is intended for the Russian market: "Buy a passport and travel everywhere."
I think the "immunity passport" will not be sufficient for COVID-19.
The re-infection of patients who recovered from COVID-19 has been reported in some studies. This suggests that - even though the production of antibodies is stimulated when one gets infected - these antibodies may not be substantially high enough to prevent a second wave of infection. A study revealed that the level of antibodies against COVID-19 in 4 out of 10 asymptomatic patients dropped to a very low level 2 months after recovery. While 1 out of 10 symptomatic patients who recovered had undetectable antibodies level 2 months after recovery. So, being exposed to the live virus or attenuated form, might not be sufficient enough to confer complete immunity as it has been shown that the level of the antibodies falls with time.