For any arbitrary NP complete language is there always a polytime superset the complement of which is also infinite?A trivial version which does not stipulate the superset to have infinite complement has been asked at https://cs.stackexchange.com/q/50123/42961For purposes of this question, you can assume that $P \ne NP$.  As Vor explained, if $P = NP$ then the answer is No.  (If $P = NP$, then $X = \{x \mid x \in \mathbb{N^+} \land x > 1\}$ is NP-complete. Clearly there is no superset of $X$ which is infinite and has an infinite complement, as the complement of $X$ has only a single element.)  Thus we can focus on the case $P \ne NP$.
Poly time superset of NP complete language with infinitely many strings excluded from it
cc.complexity theory;np complete;structural complexity
Every $\mathsf{coNP}$-complete set contains an infinite subset in $\mathsf{P}$ assuming that pseudorandom generators exist, and secure one-way permutations exist.In other words, assuming that these two conjectures are true, no $\mathsf{coNP}$-complete set is P-immune.As pointed out in the comments by Lance, this is implied by Theorem 4.4 of Glasser, Pavan, Selman, and Sengupta, Properties of NP-complete sets,SIAM J. Comput. 36(2), 516542.(Kaveh has already shown that your question is equivalent to whether every $\mathsf{coNP}$-complete sets contains an infinite $\mathsf{P}$ subset. In other language, this is saying that no $\mathsf{coNP}$-complete set is $\mathsf{P}$-immune. This is the language used in the above-referenced theorem.)