Assume that $n>1$, and that $\{p_j\}$ is a discrete sequence in $\mathbb{C}^n$ (without repetition). Is it possible to find an origin of $\mathbb{C}^n$ such that $0 < |p_1| < |p_2| < \dots$?
The answer is yes and it is used in a paper by Rosay and Rudin.
This is false. Consider the sequence in $\mathbb{C}^2$ given by $(1,0)$, $(0,0)$, $(3,0)$, $(2,0)$, $(5,0)$, $(4,0)$, ...
If $q=(x_1+iy_1,x_2+iy_2)\in\mathbb{C}^2$ and $|p_k-q|<|p_{k+1}-q|$ for all $k$, then $$|q-(2n+1,0)|^2<|q-(2n,0)|^2$$ for all $n$. This means that $$(x_1-2n-1)^2+y_1^2+x_2^2+y_2^2<(x_1-2n)^2+y_1^2+x_2^2+y_2^2,$$ which simplifies to $$-2x_1+4n+1<0.$$ But this cannot hold for all $n$.