Euler and minus sign

Something like this?

\documentclass[a4paper,12pt,fleqn]{article}
\def\imaginaryunit{j}                  % the imaginary unit, i for mathematician and theoretical physicist, j for the rest of the world.
\def\imunit{\mathrm{\imaginaryunit}}   % ... in upright math
\def\ce{\mathrm{e}}                    % the constant e, upright of course
\newcommand\epowim[1]{\ce^{\epowimaux#1\relax\endep}}
\def\epowimaux#1#2\endep{\ifx-#1\relax-\imunit\else%
  \if j\imaginaryunit\relax\,\fi\imunit#1\fi#2}
\begin{document}
\[
\epowim{\alpha}\qquad \epowim{-\alpha} \qquad \ce^{-\imunit\alpha}
\]
\[
\epowim{x+t}\quad\epowim{-x+t}\quad\epowim{-}\quad\epowim{}
\]
\def\imaginaryunit{i}
\[
\epowim{\alpha}\qquad \epowim{-\alpha} \qquad \ce^{-\imunit\alpha}
\]
\[
\epowim{x+t}\quad\epowim{-x+t}\quad\epowim{-}\quad\epowim{}
\]
\end{document}

Avoid \def, your life will be easier.

Since you seem to know about \@ifnextchar:

\documentclass[a4paper,12pt,fleqn]{article}

% the imaginary unit, j for engineers and i for the rest of the world
\newcommand\imaginaryunit{j}
% in upright type as engineers do; also Euler's constant
\newcommand\imunit{\mathrm{\imaginaryunit}}
\newcommand\ce{\mathrm{e}}

\newcommand{\fiximunit}{\if\imaginaryunit j\,\fi}

\makeatletter
\newcommand{\epowim}[1]{\ce^{\epowim@#1}}
\newcommand{\epowim@}{\@ifnextchar-{\epowim@@}{\epowim@@{\fiximunit}}}
\newcommand{\epowim@@}[1]{#1\imunit}
\makeatother

\begin{document}

\begin{equation}
\epowim{\alpha}\qquad \epowim{-\alpha} \qquad \ce^{-\imunit\alpha}
\qquad \epowim{} \qquad \epowim{-}
\end{equation}

\end{document}

Explanation: \@ifnextchar- uses the next argument if - is found, the successive one otherwise, but without removing -. So if - is called the result will be

\epowim@@-\alpha

which passes - as argument to \epowim@@. Otherwise \epowim@@ is called with \fiximunit as argument.

This would not fail with \epowim{} or \epowim{-} to typeset e^j and e^−j respectively.

A perhaps simpler implementation with xparse:

\documentclass[a4paper,12pt,fleqn]{article}

\usepackage{xparse}

% the imaginary unit, j for engineers and i for the rest of the world
\newcommand\imaginaryunit{j}
% in upright type as engineers do; also Euler's constant
\newcommand\imunit{\mathrm{\imaginaryunit}}
\newcommand\ce{\mathrm{e}}

\newcommand{\fiximunit}{\if\imaginaryunit j\,\fi}

\NewDocumentCommand{\epowim}{m}{\ce^{\powim#1}}
\NewDocumentCommand{\powim}{t-}{\IfBooleanTF{#1}{-\imunit}{\fiximunit\imunit}}

\begin{document}

\begin{equation}
\epowim{\alpha}\qquad \epowim{-\alpha} \qquad \ce^{-\imunit\alpha}
\qquad \epowim{} \qquad \epowim{-}
\end{equation}

\end{document}