Q8a: How does complex arithmetic work?  
A8a: It works mostly like regular algebra with a couple additional formulas:  
(note: a,b are reals, x,y are complex, i is the square root of -1)  
Powers of i: i^2 = -1  
Addition: (a+i*b)+(c+i*d) = (a+c)+i*(b+d)  
Multiplication: (a+i*b)*(c+i*d) = a*c-b*d + i*(a*d+b*c)  
Division: (a+i*b)/(c+i*d) = (a+i*b)*(c-i*d)/(c^2+d^2)  
Exponentiation: exp(a+i*b) = exp(a)(cos(b)+i*sin(b))  
Sine: sin(x) = (exp(i*x)-exp(-i*x))/(2*i)  
Cosine: cos(x) = (exp(i*x)+exp(-i*x))/2  
Magnitude: |a+i*b= sqrt(a^2+b^2)  
Log: log(a+i*b) = log(|a+i*b|)+i*arctan(b/a) (Note: log is multivalued.)  
Log (polar coordinates): log(r*e^(i*theta)) = log(r)+i*theta  
Complex powers: x^y = exp(y*log(x))  
DeMoivre's theorem: x^a = r^a * [cos(a*theta) + i * sin(a*theta)]  
More details can be found in any complex analysis book.  
  
Q8b: How does quaternion arithmetic work?  
A8b: Quaternions have 4 components (a+ib+jc+kd) compared to the two of  
complex numbers. Operations such as addition and multiplication can be  
performed on quaternions, but multiplication is not commutative..  
Quaternions satisfy the rules i^2=j^2=k^2=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j.  
  
See:  
  
http://www.dtek.chalmers.se/Datorsys/Project/qjulia/index.html   
QJulia page (quaternions) (Henrik Engstrvm)