Select is a verb. Selected is both its past participle and its simple past form. The simple past passive is formed with the past tense of to be and the simple past of the verb being passivised. Thus, the simple past passive of "X selects him as chairman" is
He was selected chairman
So, we can see that the principal verb of your example sentence can be selected. Could it also be was?
Was and were are used with other verbs to form the past progressive or the past simple passive. They are also used with adjectives and nouns to show that the adjective or noun applied in the past, and depending on emphasis or context it might or might not still apply now. Selected can be an adjective, but it's not usually applied to a person that way. However, it can be applied to a position:
The chairman of the organisation is a selected position
He is the selected chairman
So there is the possibility of selected not being a verb here. If it were an adjective, it would be modifying chairman, rather than being a complement itself. 'Selected chairman' would become a noun phrase.
However, if that were a noun phrase, chairman would not be a mass (uncountable) noun; it is countable, and so it needs a quantifier or article. Either a or the could be appropriate, depending on circumstance, but as an example:
He was the selected chairman.
Chairman is still a noun in the situation where selected is a verb, but in that case is referring to the position as an abstract, and so it does not require an article.
In neither case is selected a complement. It is either the principal verb, or it is an adjective applied to chairman.
It is possible to make the intention of selected as a verb more explicit with one or more extra words:
He was selected as chairman
He was selected for the role of chairman
Essentially, however, how to parse this depends on context. "At the meeting, he was selected chairman" clearly has selected as the principal verb. It is describing an action that took place at the specified time. In other contexts, it may be otherwise.