If we consider what is needed to advance science today many responses will list a number of unsolved complex mathematical problems. However these responses would presume the major hindrance to advances in science and engineering is the resolution of known, but unsolved, complex mathematical problems.  However, a more important question should be asked first: Whether our existing mathematics is adequate for modelling the wide range of phenomena we have discovered – including quantum physics, mechanics of continua, general relativity, etc.

This is a crucial question, as an inadequate mathematics will trump any list of known unsolved complex mathematical problems – in hindering the advancement of science and engineering. And resolving the ‘inadequacies’ of our current mathematics could lead to solutions to at least some of the complex mathematical problems list.

I submit that our current mathematics is inadequate for modelling the wide range of phenomena we have discovered.  Further, the specific inadequacy we need to address is our inability to represent complex values as fully formed values – without any undefined terms.

Note that our entire universe of modern science would not be possible without the decimal numeric system (and related positive-based positional systems). We could not perform our science using Roman numerals or ratios. Logarithms, scientific notation – these are not possible without this 1500+ year-old mathematical invention. The mathematics of today would not be possible without this system – including most theoretical mathematics.
This has been a crucial pre-requisite to modern science, which we have taken for granted.

Yet it has limitations. In particular it can only represent values of Real numbers (fully). As we have found increasing – even pervasive – need to include Complex numbers into our models, this Real numeric system is inadequate to (fully) representing a complex value.  We found a ‘work-around’ to representing Complex values through defining them something like ‘x + iy’ – however this is not a full representation of a Complex value precisely because it includes an undefined term i = sqrt(-1). This problem gets worse with Hamiltonian and Quaternions.

Our mathematics is inadequate and the primary area needed to resolve this issue is to define an area of mathematics currently undefined (negative-based exponents and logarithms) and to invent a new numeric system capable of fully representing Complex values.
As decimals were a pre-requisite for current Modern Science (v1.0), a new Complex-valued numeric system is a re-requisite for Modern Science and Engineering v2.0