After considering the other answers, I have some thoughts to add. These are to be taken as suggestions rather than conclusions.

 + **Simple spatial completeness** (which is usually four):
   - in one dimension (east/west in 14:8; north/south in 6:6, where "the four winds of heaven" were just mentioned in verse 5, and yet east/west are not mentioned); this could also be classified under **binary completeness** (completeness in any case of simple opposites)
   - in two dimensions (width/length in 2:2 and 5:2)
    Two angels in chapters 1 and 2
 + **Simple temporal completeness** (day/night in 14:7; winter/summer in 14:8; seems to carry the idea of eternity)
 + **Rhetorical emphasis**:
   - **Repetition**: Direct and strong (2:6; 4:7)
   - **Parallelism**: more subtle; e.g. 13:1—
     > On that day there shall be a fountain opened for the house of David and the inhabitants of Jerusalem to cleanse them from **sin** and **uncleanness**. (ESV)
 + Two of the **offices of the Messiah**, priest and king, with the third being hidden by virtue of proximity in the person of the prophet (thus the olive trees, branches, golden pipes, anointed ones, Yeshua and Zerubbabel)

That explains some of the individual cases; but is there any unifying reason why Zechariah, under the power of the Spirit of God, uses the number two? Some guesses:

 + Though we ought never to use the elegance of God's Word as a proof of its truthfulness, or confuse the beauty of the literature with the beauty of the truth, nevertheless, parts of the Scriptures have a distinct literary beauty. The use of two could be esthetic, providing a unifying literary theme.
 + The use of two through, from the first chapter to the last, could also have been intended by the Spirit of God as a rebuke to those who think that the book cannot have been written by one prophet. By providing a literary unity, as mentioned in the previous point, Zechariah's duality precludes the possibility of wanting to separate the first part of the book from his last, even though the first part of his ministry seems to have been more well received than the latter part. This is another duality within a whole, not a division.

This answer is not yet fully satisfactory to me.