Does "monotonic sequence" always mean "a sequence of real numbers" When we say a sequence is monotonic, does that imply the sequence is Real Number Sequence? And other propositions about monotonic, all real-valued?
When I see some mathematical analysis books, sometimes they talk about some properties/facts like convergence of sequences in a metric space, for example, real field and complex field, sometimes the results are the same.
However, when I see some propositions about "monotonic" are limited within real numbers.
Maybe they write for simplicity of the theorem?
For example:
Definition: A sequence $s_n$ of real numbers is said to be monotonicaly increasing if $s_n\leq s_{n+1}$.
Complex sequences may also have something monotonicity?
Basic fact I know is that we cannot compare two complex numbers, maybe we should use partial relation? norm of complex number(this is a real number)? dictionary order?

To speak of monotonicity one needs to have a notion of order. As long as the set of objects you are considering your sequences to come from is ordered in a reasonable way, you can speak of monotonicity. It is common to consider partially ordered sets, or simply posets. A poset is a pair $(S,\le )$ where $S$ is a set (which can be any set at all) and $\le $ is a transitive, reflexive, and anti-symmetric relation on $S$. In the context of posets, so for sequences of elements from a poset, monotonicity makes perfect sense. 
The real numbers are ordered by the usual meaning of $x\le y$. However, the complex numbers are not ordered in any natural useful way, so we don't speak of monotone sequences of complex numbers. An example of a poset which is useful in the context of analysis is the poset of, for instance, all functions $f:\mathbb R \to \mathbb R$. This poset is ordered by $f\le g$ precisely when $f(x)\le g(x)$ for all $x\in \mathbb R$. Then you can speak of monotone sequences of functions. 
