Should `Complex` be convertible to `Quaternion`?

[hyrodium]

We already have constructor and convert method from Complex to Quaternion.

julia> Quaternion(Complex(1,2))
Quaternion{Int64}(1, 2, 0, 0, false)

julia> convert(Quaternion, Complex(1,2))
Quaternion{Int64}(1, 2, 0, 0, false)

However, we still missing the following methods + and *.

julia> Quaternion(1,2,3,4) + Complex(1,2)
ERROR: MethodError: no method matching imag(::Complex{Int64})
You may have intended to import Base.imag
Closest candidates are:
  imag(::Quaternion) at ~/.julia/dev/Quaternions/src/Quaternion.jl:44
  imag(::Octonion) at ~/.julia/dev/Quaternions/src/Octonion.jl:48
Stacktrace:
 [1] convert(#unused#::Type{Quaternion{Int64}}, z::Complex{Int64})
   @ Quaternions ~/.julia/dev/Quaternions/src/Quaternion.jl:21
 [2] _promote
   @ ./promotion.jl:327 [inlined]
 [3] promote
   @ ./promotion.jl:350 [inlined]
 [4] +(x::Quaternion{Int64}, y::Complex{Int64})
   @ Base ./promotion.jl:379
 [5] top-level scope
   @ REPL[23]:1

julia> Quaternion(1,2,3,4) * Complex(1,2)
ERROR: MethodError: no method matching imag(::Complex{Int64})
You may have intended to import Base.imag
Closest candidates are:
  imag(::Quaternion) at ~/.julia/dev/Quaternions/src/Quaternion.jl:44
  imag(::Octonion) at ~/.julia/dev/Quaternions/src/Octonion.jl:48
Stacktrace:
 [1] convert(#unused#::Type{Quaternion{Int64}}, z::Complex{Int64})
   @ Quaternions ~/.julia/dev/Quaternions/src/Quaternion.jl:21
 [2] _promote
   @ ./promotion.jl:327 [inlined]
 [3] promote
   @ ./promotion.jl:350 [inlined]
 [4] *(x::Quaternion{Int64}, y::Complex{Int64})
   @ Base ./promotion.jl:380
 [5] top-level scope
   @ REPL[24]:1

I think we have the following choices to solve this problem.

• Remove the constructor and convert method
  • There is no natural embedding ℂ ⊆ ℍ because it depends on the choice of the basis.
  • This will make a breaking change.
• Add type promotions
  • This will not make a breaking change.

I don't know the practical benefit of converting from Complex to Quaternion, so I prefer the first.

Related comment: #29 (comment)

[sethaxen]

I agree there is not natural embedding of ℂ in ℍ, and it would be better if we did not have this conversions. Removing these would be breaking.

But this is orthogonal to the error at the beginning, which is because Quaternions has its own imag function that is different from Base.imag, and it's calling the wrong with in the convert method:

Quaternions.jl/src/Quaternion.jl

Lines 21 to 22 in 23a8212

	convert(::Type{Quaternion{T}}, z::Complex) where {T} =
	Quaternion(convert(T, real(z)), convert(T, imag(z)), convert(T, 0), convert(T, 0))

Base.imag needs to be explicitly called here. This is repaired in #56, but a separate PR to fix the bug wouldn't be a bad thing.

[dkarrasch]

Yes, there has been a quick fix for the conversion issue around in #40 (ignoring the general embedding issue).

[sethaxen]

The bug was fixed in #52