The boss is unnamed in Katherine Mansfield's story. This implies a rather imbalanced overdetermination of his bossy nature over everything else that he is--a father most importantly.
The character of the boss presents to us an illusory ego being posited and then unfixed into utter dismay through the fly episode at the end of the story. He is a pathetic character in need of a support after the untimely death of his son has killed his family-line. He faces the terror of absolute extinction and thus constructs an artifice of a stable self and mastery around him. Mr. Woodifield's articulation suddenly unmakes that protective ring and the fly as the second entrant into his snug room completes his cycle of horror by re-enacting the event of his son's death as well as implying his own death that seems to be round the corner.
As we see the boss at the end, he is reduced to a scrap, on the verge of a panic attack and already inflicted by amnesia. He has turned into his other--'old' Mr. Woodiefield.