What would the "more mathematical definition" of a vector be in your opinion? To me a vector is a special matrix which also has some special operator defined like dot product. While I would distinguish between row and column vector, in Mathcad only a column vector is named vector, a rwo vector being an 1 x n matrix.
You can represent a complex number in many different ways, a pair of real scalars similar to coordinates in 2D, component representation (a+j*b) polar representation, exponential form, versor form and then even as a 2-element vector (=2x1 matrix), but I consider the latter being quite unusal.
Anyway, to Mathcad a vector is a n*1 matrix consisting of whatever you want - real or complex scalars, strings, or even othere matrices/vectors nested in.
If you use the dot product with complex elements you have further to consider that (a,b)*(c,d) = (a*c" + b*d"), where (a,b) should denote a (column)vector and c" the complement of c. Thats the usual mathematical definition. If you expected simply a*c+b*d you would have to transopse the first vector: (a,b)T*(c,d)=a*c+b*d, thats a simple matrix multiplication.