I have seen the following conventions, e.g.
- We define $K=\mathbb C$ ...
- We define $K$ to be $\mathbb C$ ...
- We define $K:=\mathbb C$ ...
I prefer number 3, because it is concise and it is clear what is defined to be what, similarly to a programming language when one writes $K\leftarrow\mathbb C$.
But is this in fact a commonly used notation or only used by a few mathematicians?
If the sentence has the word "define" in it, then $:=$ is redundant. Redundancy isn't always bad, per se.
A mathematicians could also write it as "If $K=\mathbb C$ then $\dots$." In that sense, it isn't a definition. In computer programming, you have assignment, but you can reassign, so you get code like $x:=x+1$. That should never happen in mathematical writing, except when writing about programming/algorithms.
How do you deal with the case "$K=\mathbb C$ or $K=\mathbb R$?" That's one that often comes up, and it isn't so much a definition as a condition. And I think that's the real distinction - something what looks like a definition is often really a condition.