1 Monty: Dr Smith is nothing but a fraud! She chases all sorts of baseless myths around the world. That's not real archaeology.
2 Alice: Did she not find Atlantis?
2 Monty: Yes, but—
2 Alice: And the mystical metal orichalcum?
2 Monty: Yes, but—
3 Alice: And the Palladium of Troy?
3 Monty: Yes, but—
3 Alice: But what?
4 Monty: I was there too!
4 Alice: Criticising her methods.
4 Monty: Yes, but—

I realise Monty probably shouldn't be wearing his hat in his own office. And he should remove it when introduced to a lady. I actually considered having him without his hat for these scenes, but then I'd have to decide what colour hair he has, and, perhaps more importantly, he wouldn't look like the Monty Jones we all know.

So his hat remains on.