The spores most probably belong to a saprophytic or plant pathogenic species, which does not infect animals or humans. However, it could cause allergies (very similar looking conidia are produced by Alternaria sp., which is a very important allergenic fungus). Most fungi are not human pathogens and don't survive at body temperature (37C). However, people who are immunocompromised rather often become infected by different fungal species.
I am quite sure that these spores belong to Ulocladium, cf. U. botrytis. Ulocladium is a mould genus that grows on condensation surfaces indoors, typically in badly ventilated bedrooms. The fungus grows often on/behind wall paper on concrete outer walls during wintertime. As the wall surface is cooler than the room air, the warm and moisty air condensates on these surfaces.
The conidia is similar to the conidia of Alternaria spp. and the distrubation on the coidiaphores may help to identified. the fungus is slow grows on PDA medium and the color of mycelium is gray.
Based on this figure, Identification is difficult, But I think it belongs to the Alternaria-like species, specially Ulocladium spp. or Stemphylium spp.
The characteristics of conidia and conidiophores represent those of Ulocladium, not Pithomyces or Stemphylium. And now, Ulocladium species are re-classified as member of the genus Alternaria section Ulocladium.
Seems Ulocladium. Alternaria has conidia forming chains (as well as Pithomyces) but not Ulocladium. Be sure that conidia chains are not evident to discard Alternaria. In addition Ulocladium has geniculata conidiophore i.e. strongly bent but not Alternaria.
I vote for Ulocladium as several people said already.