The Wikipedia page gives the intersection of the set with the real axis as $[-2,0.25]$
Added: You can verify that $-2$ is in the set easily, and that any more negative number decreases each iteration without bound. For the positive end, each iteration is greater than the one before. To hit a limit, you must have $z=z^2+c$, which has the solution $z=\frac{1+\sqrt{1-4c}}2$, which becomes imaginary at $c \gt \frac 14$
You ask: "Is Wikipedia's definition of fractal the standard?" and right near the top of Wikipedia's page of fractals, we see the following definition:
A fractal is a mathematical set that has a fractal dimension that
usually exceeds its topological dimension and may fall between the
integers.
The statement that the fractal dimension may "fall between the integers" really adds nothing but, other than that, I would say that this is fairly standard; it is unquestionably the definition that was put forward by Mandelbrot around 1975 when he coined the term "fractal". He did not refer to "fractal dimension" at that time but, rather, the "Hausdorff-Besicovitch dimension" as he put it. In fairness, the usefulness of this definition has been debated with even Mandelbrot himself feeling that it might not be inclusive enough. Nonetheless, this comparison of dimension is central in fractal geometry. Gerald Edgar calls his great book, Measure, Topology, and Fractal Geometry, a meditation on the definition.
Taking this to be the definition, we can definitely say that the Cantor set satisfies it. If by "fractal dimension" you mean similarity dimension, then the Cantor set has fractal dimension $\log(2)/\log(3)$, since it's composed of two copies of itself scaled by the factor three. Also, the set is regular enough that any reasonable definition of fractal dimension agrees with that computation. (Well, any real-valued defintion.)
Topological dimension is a trickier thing, actually. It's inductive in nature. Totally disconnected sets (like single points, finite sets, or notably the Cantor set) have dimension zero. Higher dimensions are defined in terms of lower dimensions. The space we live in is three dimensions because balls in this space have a surface that is two dimensional. Because of this inductive nature, topological dimension always yields an integer.
When you write that you "do not see the irregular aspects or the complexity that is usually inherent with fractals", I think you might have a bit of a mis-understanding about fractal geometry. The Cantor set is indeed regular but, then so are all the strictly self-similar sets studied in classical fractal geometry - the Koch curve, the Sierpinski triangle, the Menger sponge, and countless others all display this regularity. Indeed, it's exactly this regularity that allows us to understand them.
To emphasize this regularity, and how it appears in not just the Cantor set, compare the following zooms of
The Cantor set
The Koch curve
Now, of course, there are "irregular" fractals - or, at least, less regular fractals. Examples include random version of self-similar sets, examples that arise from number theory, and examples arising from complex dynamics (like Julia sets). It's not their irregularity that makes these objects fractal, however. On the contrary, its the regularity that we can find that allows us to analyse these objects to the point where we can characterize them as fractal. Of course, this analysis is bit harder with these less regular examples.
Best Answer
First, a distinction should be made: a fractal is one thing, and certain methods for constructing particular fractals are another.
Loosely, a fractal can be described as an object which is self similar at different scales, that is, "zooming in" repeatedly leads to the same curve. An interesting property of fractals which is sometimes used to define them, is that one can assign a non integer dimension to them. For instance, a smooth curve has dimension 1, but a Koch snowflake is in a certain sense closer to being a two dimensional object, and we can assign it a non integer dimension of $\sim1.26$. Intuitively, A Sierpinski carpet is even closer to a 2D object, and indeed we assign it a higher fractal dimension, of $\sim1.89$.
As to Mandelbrot's famous set, the idea is as follows: To check whether or not a (complex valued) point $c$ is in the set, start with $z_0=0$, and iterate. When the series stays bound, $c$ is in the set. When the series diverges, $c$ is not in the set. (try $c=-1,0,1$ for yourself, and see what you get). For instance, is the point $i$, i.e. $(0,1)$ in the Mandelbrot set? $$z_1 = 0^2 + i = i,\quad z_2 = i^2 + i = i - 1,\quad z_3 = (i-1)^2+i= -i$$ $$z_4 = (-i)^2+i = i-1$$ Thus, the point $i$ leads to a bound repeated loop, and is therefore in the Mandelbrot set (i.e. the black area in most drawings).