For one thing, some languages apparently do not have a way to coordinate two noun phrases on an equal level (or at least, no documented way). See this question: What is the difference between AND and WITH in general? and the related WALS article Noun Phrase Conjunction, by Leon Stassen.
The basic distinction is between those languages which use a different marker for noun phrase conjunction and comitative phrases (so-called and-languages; Stassen 2000) and those languages in which the markers for noun phrase conjunction and comitative phrases are the same (with-languages). A clear example of an and-language is English. As the examples in (1) demonstrate, in this language the marker of noun phrase conjunction (and) is different from the marker of comitative phrases (with). In contrast, the example in (2) shows that Nkore-Kiga (Bantu; Uganda) is a with-language: the item na can be translated as either 'and' or 'with'. An alternative way of stating this is to say that with-languages like Nkore-Kiga lack the structural option of noun phrase conjunction.
a. John and Mary went to the movies.
b. John went to the movies with Mary.
(2) Nkore-Kiga (Taylor 1985: 58)
n-ka-za-yo na Mugasho
1sg-rec.pst-go-there and/with Mugasho
‘Mugasho and I went there./ I went there with Mugasho.’
So apparently, if you translate ‘Mugasho and I went there’ into Nkore-Kiga, you'd have to use a singular verb. It's unclear if this really satisfies the criterion of a "complex subject", though.
In any case, languages that exclusively use this type of strategy are apparently uncommon and they seem to be prone to developing the option of using plural (or dual etc. as appropriate) agreement:
Although both for with-languages and for and-languages clear instances can be found, the typological status of these two types is probably not equally well established. In particular, there is a notable discrepancy in the stability of the types. In general, and-languages can be said to be stable diachronically and “pure” in their synchronic state: there is a sharp delineation between the two available strategies. On the other hand, “pure” instances of with-languages are relatively rare. For a considerable number of such languages, some process of “diachronic drift” of the comitative encoding can be attested. The general outcome of this process is to effectuate a shift from one-strategy to two-strategy status. [...] Among the possible “paths” involved in the shift, we encounter cases in which
(ii) There is differentiation in person/number agreement (mainly between singular and dual/plural). An example is Tolai (Oceanic; New Britain, Papua New Guinea).
(11) Tolai (Mosel 1984: 176)
a. nam ra tutana i ga rovoi ma ra pap
DEM ART man 3SG PFV hunt with ART dog
‘That man went hunting with his dog.’
b. Terengai dir rovoi ma ra pap
Terengai 3DU hunt and/with ART dog
‘Terengai and his dog went hunting.’
Distinct from the and/with issue, some languages seem to use proximity agreement at least more frequently than English, even if they don't use it in all cases.
"Conjunctions and Grammatical Agreeement", by Heidi Lorimor (2001), says
Another type of flexibility observed with conjoined subjects is the
option of single conjunct agreement, in which verbs and predicate
adjectives agree with only one of the two conjuncts. The most famous
case of single conjunct agreement is in Modern Standard Arabic, where
agreement with the closest conjunct is obligatory when the verb
precedes the subject (Aoun, Benmamoun, & Sportiche, 1994). (2)
However, in MSA verbs preceding their subjects are always singular, regardless of the number of the subject, so this proximity agreement never actually shows up in verbal number marking, only in verbal gender marking and pronominal gender and number marking (3). So MSA verb agreement does not seems to be a good example for your question, but the phenomenon it illustrates may show up in other languages that do have verbal number agreement in this position:
This is alternatively termed “Partial Agreement”, “Closest Conjunct
Agreement”, or “First Conjunct Agreement”, because this type of
agreement occurs most often when the verb precedes the subject, where the first conjunct is the closest conjunct (Corbett, 2006). To illustrate, if English had First Conjunct Agreement with postverbal subjects, as some have argued for “there” constructions (Munn, 1999), you would see a singular verb, followed by a conjoined subject (e.g., “There is a dog and a cat in the room”), if the first conjunct is singular. In a language with gender agreement, the verb would agree with the first (closest) conjunct in gender and number, as if the first conjunct were the full subject of the verb.
This phenomenon is most common with postverbal conjoined subjects, where agreement is with the first (closest) conjunct. It is, however, also documented with the closest (second) conjunct and with the furthest (first) conjunct for preverbal subjects in languages such as Latin and Slovene (Badecker, 2007). The pattern that has never been observed, however, is furthest (second) conjunct agreement with postverbal subjects (Corbett, 2006). (2)
If you do a Google search for "single conjunct agreement", "closest conjunct agreement" or "first conjunct agreement", there are a number of results that give more relevant examples, but my impression so far is that all of the examples seem to be from languages that also have some kind of plural agreement as an option in at least some cases.