Give an example of a function φ:C→C such that
φ(w+z)=φ(w)+φ(z)
for all w,z∈C but φ is not linear. (Here C is thought of as a complex vector space.)
If φ is linear then φ(1z)=zφ(1) meaning every value of φ is determined from φ(1). We just need to construct a φ that doesn't have φ(z)=zφ(1) but is still additive.
Define φ(1)=1, additivity immediately implies φ(r)=r for all rational r (see Additive Domain Extension).
Additivity doesn't give us a value for φ(i) though, meaning we're free to define it as φ(i)=0. φ is clearly still additive, but
φ(i)=0=iφ(1)=i
If you prefer to see φ defined all at once here you go
φ(a+bi)={a0if b=0if a=0
The reason it's hard to find φ:R→R is we can show φ(rx)=rφ(x) for all rational r, meaning we have to somehow squeeze nonlinearity in using irrational scalars!