In what sense can a complex number be a scalar? A definition of a scalar like

A scalar is a one-component quantity that is invariant under rotations of the coordinate system (see http://mathworld.wolfram.com/Scalar.html)

seems to exclude complex numbers from being scalars, because complex numbers are generally expressed as two-component vector-like quantities. However in physics we have things like complex scalar fields. In what sense can complex numbers be scalars in physics? Does it mean, that a scalar is defined in physics simply as any quantity (independent of the number of components), that is invariant under space-time transformations (translations, rotations and Lorentz boosts)?
 A: To amplify WetSavannaAnimal's answer a bit, a mathematician defines a vector space (loosely) as a set of things that behave like little arrows when added together or multiplied by a scalar (aka number). They need not be little arrows. E.G. The set of all functions $y = ax^2 + bx + c$ is a 3D vector space.
An n dimensional vector can always be represented by n numbers, which is equivalent to a point in an n dimensional physical space, or a little arrow from the origin to that point. This is the sense in which a vector can be described by a magnitude and direction. 
For the most familiar vector spaces, the numbers are real. But it is possible for them to be complex as well. E.G. the functions above could be defined over the complex plane. It would still be a 3D vector space. Even though $a$, $b$, and $c$ would be complex numbers, there are 3 of them. 
This stretches the idea of an arrow in a physical space a bit. But then, so does a 4D or 17D vector. The point is that a scalar is the number that can multiply a vector without changing its direction. 
To a physicist, a vector has to have another property. It must have a physically meaningful magnitude that does not change when you rotate the coordinate system. To a physicist, force is a vector, but a point in a thermodynamic phase space is not. To a physicist, 4D space-time is a vector space where the magnitude is the interval and the coordinate rotations are boosts. 
Physicists are a bit sloppy on this point. To a mathematician, the idea of magnitude is captured by the definition of a norm. To a mathematician, 4D space-time is not a normed vector space because a norm must never be negative. 
Getting back to the point, a second meaning of scalar is a physically meaningful value that is invariant under coordinate rotations. The magnitude of a vector is a scalar. Likewise, magnitudes of higher rank tensors are scalars. 
In this sense, scalars are usually real numbers. Quantum mechanics has complex valued wave functions. But the physically meaningful magnitudes are real. 
A: What you've quoted is a definition of a "scalar" in some physical/mathematical context.
The term "scalar" comes from the Latin word scala meaning ladder; and multiplying a vector quantity by a scalar has the effect of scaling its magnitude without affecting its orientation. Hence the name "scalar". But over the years, "scalar" has gradually been bastardized by mathematicians to even refer to complex quantities "scaling" some other abstract mathematical quantity via multiplication. Despite the fact that originally, multiplying a vector by a complex quantity had the effect of both scaling and rotating a vector!
So a complex number can be a scalar today when it's used to "scale" another mathematical abstract quantity via the unary operation we call multiplication. But in a way that wasn't originally intended through the definition of a "scalar".
A: 
A scalar is a one-component quantity that is invariant under rotations of the coordinate system

OK, but what then do you mean by "rotation"?
See, a scalar in the sense as defined in your quote is not just "a scalar", period. You can only have a scalar with respect to some particular rotation operation. The same quantity can be a scalar with respect to one kind of rotation and a vector or tensor with respect to another.
It's true that there is a rotation group (a $U(1)$ group) which acts on the complex plane and turns one complex number into another. But that's not the type of rotation group physicists use. We use rotations that turn physical directions into one another (the traditional $SO(3)$ rotation), or that turn worldline directions into one another (the Lorentz group), or that turn spin states into one another (any $SU(2)$ spin group), or color states (the $SU(3)$ group used in QCD), or so on. None of these rotations affects a plain old complex number, because a plain old complex number doesn't have any physical meaning attached that would cause it to change under any physical rotation operation.
This has implications for what counts as a "component". As user2357112 mentioned in the comments, it depends on context: for example, you can treat a complex number as a two-component vector, or you could have a vector with complex coefficients (as in quantum mechanics), in which case each complex number is only one component. In fact, there are even situations where an entire matrix can be a component, such as the Pauli vector.
The point is that you shouldn't assume a component has to be a real number, or even any sort of number. It probably makes more sense to define a component in terms of rotations (since in math the whole idea of components comes from vector spaces, so we might as well do the analogous thing in physics). I'm not going to suggest any sort of rigorous definition here, but a sensible one would capture the idea that components of a vector "trade off" among each other under a rotation, and if some mathematical object isn't affected by a certain rotation, then the whole object (whether it's number, vector, tensor, whatever) deserves to be considered one component (and thus a scalar) with respect to that rotation.
A: Although this is kind of trivial, a complex number, as a member of a field can be a scalar that acts by commutative multiplication on a vector space, the latter, through scaling, being the fundamental manifestation of the the notion of linearity. See the definition of a vector space for more details.
A: Complex numbers are usually visualized as a "two-component vector-like quantity". However, this is just a visualization tool, and the real+imaginary axes of the Argand plane do not correspond to any physical directions. Complex numbers do not change under $SO(3)$ rotations of space or Lorentz boosts, which is why they are scalars.
If you think complex numbers are fundamentally linked to points on a 2-D surface, you might be interested in their history. Many important theorems about complex numbers were developed in the 18-th century, including de Moivre's formula and Euler's formula. All of these were based on the algebraic definition $i^2 = -1$, without any geometric identification/visualization of complex numbers as points in a complex plane. It was only in the 19-th century that the complex plane was born as a concept.
