Hi,
electromagnetic waves have always vector character and a common relation (yield from Maxwell's equations) between the electric field \vec E(\vec r,t) and the magnetic field \vec B(\vec r,t) is
\vec B(\vec r,t) = \frac{1}{\omega} \, \vec k \times \vec E(\vec r,t)
As you stated waves can have a more complex form! The most general form can be represented by the Fourier transformation and is a consequence of the linearity of Maxwell's equations. A general expression for the electric field in this form is
\vec E(\vec r, t) = \frac{1}{(2\pi)^{3/2}} ~ \int ~ \mathrm{d}^3k ~ \tilde{ \vec E}(\vec k) \, e^{i(\vec k \vec r - c |\vec k|t)}
where \tilde{\vec E}(\vec k) is the amplitude of a plane wave that belongs to the wavevector \vec k. This is often called a wave packet.
So we are talking about the amplitudes of the electric field belonging to the plane wave with wavevector \vec k. But what is about the amplitudes of the magnetic field?
For the amplitudes the first equation is always true, because \tilde{\vec B}, \tilde{\vec E}, \vec k are always perpendicular. So the amplitudes of the magnetic field are
\tilde{\vec B}(\vec k) = \frac{1}{\omega} \vec k \times \tilde{\vec E}(\vec k)
So \vec B(\vec r,t) is analogous to \vec E(\vec r,t)
\vec B(\vec r, t) = \frac{1}{(2\pi)^{3/2}} ~ \int ~ \mathrm{d}^3k ~ \tilde{\vec B}(\vec k) \, e^{i(\vec k \vec r - c |\vec k|t)} = \frac{1}{(2\pi)^{3/2}} ~ \int ~ \mathrm{d}^3k ~ \, \frac{1}{\omega} \, \vec k \times \tilde{\vec E}(\vec k) ~ e^{i(\vec k \vec r - c |\vec k|t)}
So, the above equation is really the general solution of Maxwell's equations! Like you have seen, nothing is as easy as it seems (remember plane waves aren't physical but the infinite sum of them are as long as they vanish in infinty).
Hope i could help...
