To nit-pick on your question, scalars could be anything that you can define a field with. As long as you also define addition, subtraction, commutative multiplication, and division without zero, your scalars might be rabbits, or anything. Of course, this is completely equivalent to choosing numbers instead.
The concept of a "number" extends to hypercomplex numbers like quaternions or octonions. However, their multiplication is not commutative, and hence they do not form a field. A vector space requires an underlying field. Multiplication of integer, rational, real, or complex numbers satisfy commutativity. This, among other requirements, makes them candidates for fields.
Refer to the Wikipedia article on fields to get an impression of more possibilities, most notably finite fields.
In the context of modules (a generalized notion of vector spaces), a scalar is not required to form a field, but only a ring instead, i.e., division need not be defined. This includes quaternions as scalars, in this context (!). Considering the fact that you delve in quantum mechanics, you might want to extend these notions to octonions as well. Many arithmetic and algebraic lemmata also apply to these.