Is complex valued function like $y(t) = t^2 + i\cdot t^2$ a periodic function?
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You can simply factorize your function as $y(t)=(1+\mathrm i)t^2$. So the question boils down to whether $t^2$ is a periodic function. That would mean that there's an $a\ne 0$ so that for all $t$, $(t+a)^2 = t^2$. Now applying the binomial formula and subtracting $t^2$ on both sides gives the equation $2ta+a^2=0$ or, if $a\ne 0$, $t=-a/2$, which clearly cannot be true for arbitrary $t$. Therefore the function is not periodic.
No non-constant polynomial $p(x)$ can ever be periodic. If it were, there would be infinitely many solutions to $p(x)-p(0) = 0$.