  We employ the generalized Remez algorithm, initially suggested by P. T. P.
Tang, to perform an experimental study of Chebyshev polynomials in the complex
plane. Our focus lies particularly on the examination of their norms and zeros.
What sets our study apart is the breadth of examples considered, coupled with
the fact that the degrees under investigation are substantially higher than
those in previous studies where other methods have been applied. These
computations of Chebyshev polynomials of high degrees reveal discernible
patterns which allow for conjectures to be formulated based on abundant
experimental evidence. The use of Tang's algorithm allows for computations
executed with precision, maintaining accuracy within quantifiable margins of
error. Additionally, as a result of our experimental study, we propose what we
believe to be a fundamental relationship between Chebyshev and Faber
polynomials associated with a compact set.
