Nobody knows. It is known that we live in three measurements: it is length, width and height.
In certain cases for the description of difficult processes, it is applied the bigger number of measurements of space is considerable: four, six, ten, hundred, one thousand etc.
However, there is a contradiction in physical and geometrical measurements.
In the main classical cases great researchers manage only one measurement.
So, de Broglie's wave has only one measurement – length. But after all any physical subject has to have except length also height.
The norm (squared) of the wave function at a point x |\psi(x)|^2 is the probability density to find a particle at x, so yes the wave function has a "height".
I do not think the problem is well-posed. You mean the de Broglie wave has only one dimension-length x. However it could be of another dimensional as long as one reconsiders the problem (equivalently the wave equation) in 2D or 3D case. This has nothing to do with classical vs quantum etc.
Indeed, the question does not appear to make much sense, and if "nobody knows" why bother asking? In any case: waves have amplitude (height?), here a complex number, they also have phase, wavelength, velocity, and direction. Plenty of properties, not just "one measurement".
I am grateful to colleagues for a set of answers diversely on this matter, especially, for the answer that the size of "heights" of a wave is proportional to a square of its length.
The following paper gives an overview of de Broglie's work:
N. Philippot, Louis de Broglie:
http://home.cc.umanitoba.ca/~loly/deBroglie.pdf
Perhaps the followers of this thread will find the following book useful:
B. Cox, J. Forshaw, The Quantum Universe, 2011:
http://atibook.ir/dl/en/Siences/Natural%20sciences/physic/9780306819643_the_quantum_universe.pdf
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