As I noted in my first comment, I think this will work:

Base.sum(v::AbstractArray{<:Complex}, w::AbstractWeights) = dot(w, v)

For the general case, it looks like broadcasting is slower than the dot fallback:

julia> x = rand(10_000);

julia> z = Vector{Union{Float64, Missing}}(x);

julia> @btime dot(x, z);
  13.443 μs (1 allocation: 16 bytes)

julia> f(A, w) = sum(Base.Broadcast.Broadcasted(*, (A, w)))
f (generic function with 1 method)

julia> @btime f(x, z);
  22.258 μs (11 allocations: 240 bytes)

Since we don't actually need the broadcasting features, I guess we should simply use a loop similar to dot, but without calling dot recursively:

function dot(x::AbstractArray, y::AbstractArray)
    lx = length(x)
    if lx != length(y)
        throw(DimensionMismatch("first array has length $(lx) which does not match the length of the second, $(length(y))."))
    end
    if lx == 0
        return dot(zero(eltype(x)), zero(eltype(y)))
    end
    s = zero(dot(first(x), first(y)))
    for (Ix, Iy) in zip(eachindex(x), eachindex(y))
        @inbounds s += dot(x[Ix], y[Iy])
    end
    s
end

Wrong weighted sum and mean results for complex numbers #518

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