Zener diode BZX55C3V3 does not behave as Zener diode
That looks entirely normal to me. 3.3V zeners have a very soft knee and are unsuitable for this purpose. The zener voltage is guaranteed to be between 3.1V and 3.5V at 5mA test current, which agrees with your number with an 8V input (3.2 \$\le\$ 3.4 \$\le\$ 3.5V) with zener current approximately 4.6mA. In other words, it's behaving exactly like a 3.3V zener diode, which in this case is not good at all. 

A diode to a clamp will do better, but without more information it's hard to make recommendations. A TL431 with a 1mA current (from a 5V circuit) and a diode to your ADC input should work, but you'll have to keep the clamp voltage to maybe 3V, and some current still will flow into the ADC input. A R-R IO op-amp will clamp it more positively. 

If your actual input voltage never exceeds 1 volt, the datasheet I linked above guarantees that the current at 25°C will not exceed 2uA, so your error will be less than 2mV, but at 125°C it could be as bad as 40mV error. Chances are you want to use more than the bottom 1/3 of your ADC range. 

Edit:- 

"Zener" diodes of greater than about 5.6V are actually avalanche diodes, and they have a much sharper knee and a positive tempco. Here is a set of curves from a totally different family of small zener diodes (lifted from a 20-year-old paper Toshiba databook). As you can see, at about 8.2V and above, they are really, really good, with little voltage change from 1uA to 10mA. The "3.3V" one changes from about 1.25V to about 3V. This is a consequence of the physics involved, and you'll find some zeners shift the curves up and down a bit, but the shapes will be similar. Active circuits are required to do much better.  



Unfortunately, for some reason, this kind of info is often omitted from modern datasheets, even though it was considered important enough to murder trees for in the old days.