Construct a PDA for the complement of $a^nb^nc^n$
No, this is context-free. To accept $a^nb^nc^n$, you need to make sure that three numbers are equal. To accept $a^*b^*c^* \setminus a^nb^nc^n$, you just need to make sure that you're in one of the following three cases:


The number of $a$s is different from the number of $b$s; or
The number of $a$s is different from the number of $c$s; or
The number of $b$s is different from the number of $c$s.


Write a PDA for each of these cases, then combine them by jumping nondeterministically to each one from the start state.