Q:

Tikz: Aligning origins of two figures

I create two graphics but the vertical positions of the origins are not same.
How can we set them to have the same vertical level.
Here is my MWE. Also if you have any other comments for the plots, they are welcome.
\documentclass[12pt,reqno]{article}

\usepackage{amsfonts}
\usepackage[top=20mm,bottom=20mm,left=20mm,right=20mm]{geometry}
\usepackage{graphicx}
\usepackage{float}
\usepackage{subfig}

\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\definecolor{mc1}{rgb}{0.368417,0.506779,0.709798}
\definecolor{mc2}{rgb}{0.880722,0.611041,0.142051}
\newcommand{\R}{\mathbb{R}}
\begin{document}

\begin{figure}[ht]
  \centering
    \subfloat[The surface $z=f(x,y)$.]{\label{sespex2fig1a}
      \begin{tikzpicture}[declare function={sgn(\x)=(\x>=0)*1+(\x<=0)*(-1);f(\x,\y)=(\x<=0)*(-\y)+(\x>=0)*((abs{\y}>=\x^2)*(sgn(\y)*(sqrt{abs{\y}}+1)*\x-\y)+(abs{\y}<=\x^2)*(sgn(\y)*sqrt{abs{\y}}));}]
        \begin{axis}[
          width=100mm,
          height=60mm,
          view={120}{75},
          axis lines=center,
          xtick={-1,0,1},
          x tick label style={opacity=0.4},
          xmin=-1.25,
          xmax=1.25,
          domain=-1:1,
          xlabel={$t$},
          xlabel style={opacity=0.4},
          x axis line style={opacity=0.6},
          ytick={-1,0,1},
          y tick label style={opacity=0.4},
          ymin=-1.25,
          ymax=1.25,
          y domain=-1:1,
          ylabel={$y$},
          ylabel style={opacity=0.4},
          y axis line style={opacity=0.6},
          %ztick={-1,1},
          zticklabels={\empty},
          z tick label style={opacity=0.4},
          zmin=-0.75,
          zmax=0.75,
          zlabel={$z$},
          zlabel style={opacity=0.4},
          z axis line style={opacity=0.6},
        ]
          \addplot3[surf,color=mc1,opacity=0.2,samples=25,samples y=25,faceted color=mc1!40]{f(x,y)};
          \draw[variable=\t,domain=-1:1,samples=2,color=mc2,thick,dashed] plot (axis cs:0,\t,-\t);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,\t^2,\t);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,-\t^2,-\t);
        \end{axis}
      \end{tikzpicture}
    }
    \qquad
    \subfloat[The parses of the domain of $f$.]{\label{sespex2fig1b}
      \begin{tikzpicture}
        \begin{axis}[
          width=100mm,
          height=60mm,
          view={0}{90},
          axis lines=center,
          xtick={-1,0,1},
          x tick label style={opacity=0.4},
          xmin=-1.25,
          xmax=1.25,
          domain=-1:1,
          xlabel={$t$},
          xlabel style={opacity=0.4},
          x axis line style={opacity=0.6},
          ytick={-1,0,1},
          y tick label style={opacity=0.4},
          ymin=-1.25,
          ymax=1.25,
          y domain=-1:1,
          ylabel={$y$},
          ylabel style={opacity=0.4},
          y axis line style={opacity=0.6},
          ztick={-1,1},
          zticklabels={\empty},
          z tick label style={opacity=0.4},
          zmin=-1.25,
          zmax=1.25,
          zlabel={$z$},
          zlabel style={opacity=0.4},
          z axis line style={opacity=0.6},
        ]
          \addplot3[surf,color=mc1,opacity=0.2,samples=25,samples y=25,faceted color=mc1!40]{0};
          \draw[variable=\t,domain=-1:1,samples=2,color=mc2,very thick,dashed] plot (axis cs:0,\t,0);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,\t^2,0);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,-\t^2,0);
        \end{axis}
      \end{tikzpicture}
    }
    \caption{Graphics of $f:\R^{2}\to\R$.}\label{sespex2fig1}
\end{figure}

\end{document} 

A:

see, which one below figures better fit to your expectation:

for second figure I enlarge height of the firs sub image;
\documentclass[12pt,reqno]{article}

\usepackage{amsfonts}
\newcommand{\R}{\mathbb{R}}
\usepackage[top=20mm,bottom=20mm,left=20mm,right=20mm]{geometry}
\usepackage{graphicx}
\usepackage{float}
\usepackage{subfig}

\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\definecolor{mc1}{rgb}{0.368417,0.506779,0.709798}
\definecolor{mc2}{rgb}{0.880722,0.611041,0.142051}

\begin{document}

\begin{figure}[ht]
  \centering
    \subfloat[The surface $z=f(x,y)$.]{\label{sespex2fig1a}
      \begin{tikzpicture}[baseline=(current bounding box.center),% <---- added
      declare function = {sgn(\x)=(\x>=0)*1+(\x<=0)*(-1);f(\x,\y)=(\x<=0)*(-\y)+(\x>=0)*((abs{\y}>=\x^2)*(sgn(\y)*(sqrt{abs{\y}}+1)*\x-\y)+(abs{\y}<=\x^2)*(sgn(\y)*sqrt{abs{\y}}));}]
        \begin{axis}[
          width=100mm,
          height=60mm,
          view={120}{75},
          axis lines=center,
          xtick={-1,0,1},
          x tick label style={opacity=0.4},
          xmin=-1.25,
          xmax=1.25,
          domain=-1:1,
          xlabel={$t$},
          xlabel style={opacity=0.4},
          x axis line style={opacity=0.6},
          ytick={-1,0,1},
          y tick label style={opacity=0.4},
          ymin=-1.25,
          ymax=1.25,
          y domain=-1:1,
          ylabel={$y$},
          ylabel style={opacity=0.4},
          y axis line style={opacity=0.6},
          %ztick={-1,1},
          zticklabels={\empty},
          z tick label style={opacity=0.4},
          zmin=-0.75,
          zmax=0.75,
          zlabel={$z$},
          zlabel style={opacity=0.4},
          z axis line style={opacity=0.6},
        ]
          \addplot3[surf,color=mc1,opacity=0.2,samples=25,samples y=25,faceted color=mc1!40]{f(x,y)};
          \draw[variable=\t,domain=-1:1,samples=2,color=mc2,thick,dashed] plot (axis cs:0,\t,-\t);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,\t^2,\t);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,-\t^2,-\t);
        \end{axis}
      \end{tikzpicture}
    }
    \qquad
    \subfloat[The parses of the domain of $f$.]{\label{sespex2fig1b}
      \begin{tikzpicture}[baseline=(current bounding box.center)]% <---- added
        \begin{axis}[
          width=100mm,
          height=60mm,
          view={0}{90},
          axis lines=center,
          xtick={-1,0,1},
          x tick label style={opacity=0.4},
          xmin=-1.25,
          xmax=1.25,
          domain=-1:1,
          xlabel={$t$},
          xlabel style={opacity=0.4},
          x axis line style={opacity=0.6},
          ytick={-1,0,1},
          y tick label style={opacity=0.4},
          ymin=-1.25,
          ymax=1.25,
          y domain=-1:1,
          ylabel={$y$},
          ylabel style={opacity=0.4},
          y axis line style={opacity=0.6},
          ztick={-1,1},
          zticklabels={\empty},
          z tick label style={opacity=0.4},
          zmin=-1.25,
          zmax=1.25,
          zlabel={$z$},
          zlabel style={opacity=0.4},
          z axis line style={opacity=0.6},
        ]
          \addplot3[surf,color=mc1,opacity=0.2,samples=25,samples y=25,faceted color=mc1!40]{0};
          \draw[variable=\t,domain=-1:1,samples=2,color=mc2,very thick,dashed] plot (axis cs:0,\t,0);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,\t^2,0);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,-\t^2,0);
        \end{axis}
      \end{tikzpicture}
    }
    \caption{Graphics of $f:\R^{2}\to\R$.}
    \label{sespex2fig1}
\end{figure}

\begin{figure}[ht]
  \centering
    \subfloat[The surface $z=f(x,y)$.]{\label{sespex2fig1a}
      \begin{tikzpicture}[baseline=(current bounding box.center),% <---- added
      declare function = {sgn(\x)=(\x>=0)*1+(\x<=0)*(-1);f(\x,\y)=(\x<=0)*(-\y)+(\x>=0)*((abs{\y}>=\x^2)*(sgn(\y)*(sqrt{abs{\y}}+1)*\x-\y)+(abs{\y}<=\x^2)*(sgn(\y)*sqrt{abs{\y}}));}]
        \begin{axis}[
          width=100mm,
          height=84mm,
          view={120}{75},
          axis lines=center,
          xtick={-1,0,1},
          x tick label style={opacity=0.4},
          xmin=-1.25,
          xmax=1.25,
          domain=-1:1,
          xlabel={$t$},
          xlabel style={opacity=0.4},
          x axis line style={opacity=0.6},
          ytick={-1,0,1},
          y tick label style={opacity=0.4},
          ymin=-1.25,
          ymax=1.25,
          y domain=-1:1,
          ylabel={$y$},
          ylabel style={opacity=0.4},
          y axis line style={opacity=0.6},
          %ztick={-1,1},
          zticklabels={\empty},
          z tick label style={opacity=0.4},
          zmin=-0.75,
          zmax=0.75,
          zlabel={$z$},
          zlabel style={opacity=0.4},
          z axis line style={opacity=0.6},
        ]
          \addplot3[surf,color=mc1,opacity=0.2,samples=25,samples y=25,faceted color=mc1!40]{f(x,y)};
          \draw[variable=\t,domain=-1:1,samples=2,color=mc2,thick,dashed] plot (axis cs:0,\t,-\t);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,\t^2,\t);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,-\t^2,-\t);
        \end{axis}
      \end{tikzpicture}
    }
    \qquad
    \subfloat[The parses of the domain of $f$.]{\label{sespex2fig1b}
      \begin{tikzpicture}[baseline=(current bounding box.center)]% <---- added
        \begin{axis}[
          width=100mm,
          height=60mm,
          view={0}{90},
          axis lines=center,
          xtick={-1,0,1},
          x tick label style={opacity=0.4},
          xmin=-1.25,
          xmax=1.25,
          domain=-1:1,
          xlabel={$t$},
          xlabel style={opacity=0.4},
          x axis line style={opacity=0.6},
          ytick={-1,0,1},
          y tick label style={opacity=0.4},
          ymin=-1.25,
          ymax=1.25,
          y domain=-1:1,
          ylabel={$y$},
          ylabel style={opacity=0.4},
          y axis line style={opacity=0.6},
          ztick={-1,1},
          zticklabels={\empty},
          z tick label style={opacity=0.4},
          zmin=-1.25,
          zmax=1.25,
          zlabel={$z$},
          zlabel style={opacity=0.4},
          z axis line style={opacity=0.6},
        ]
          \addplot3[surf,color=mc1,opacity=0.2,samples=25,samples y=25,faceted color=mc1!40]{0};
          \draw[variable=\t,domain=-1:1,samples=2,color=mc2,very thick,dashed] plot (axis cs:0,\t,0);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,\t^2,0);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,-\t^2,0);
        \end{axis}
      \end{tikzpicture}
    }
    \caption{Graphics of $f:\R^{2}\to\R$.}
    \label{sespex2fig1}
\end{figure}

\end{document}

addendum: considering John Kormylo comments below, you can for the firs figure write:
\documentclass[12pt,reqno]{article}

\usepackage{amsfonts}
\newcommand{\R}{\mathbb{R}}
\usepackage[top=20mm,bottom=20mm,left=20mm,right=20mm]{geometry}
\usepackage{graphicx}
\usepackage{float}
\usepackage{subfig}

\usepackage{pgfplots}
\pgfplotsset{compat=1.15}

\definecolor{mc1}{rgb}{0.368417,0.506779,0.709798}
\definecolor{mc2}{rgb}{0.880722,0.611041,0.142051}

\begin{document}

\begin{figure}[ht]
  \centering
    \subfloat[The surface $z=f(x,y)$.]{\label{sespex2fig1a}
      \begin{tikzpicture}[baseline=(origin),% <---- suggestion of John Kormylo
      declare function = {sgn(\x)=(\x>=0)*1+(\x<=0)*(-1);f(\x,\y)=(\x<=0)*(-\y)+(\x>=0)*((abs{\y}>=\x^2)*(sgn(\y)*(sqrt{abs{\y}}+1)*\x-\y)+(abs{\y}<=\x^2)*(sgn(\y)*sqrt{abs{\y}}));}]
        \begin{axis}[
          width=100mm,
          height=60mm,
          view={120}{75},
          axis lines=center,
          xtick={-1,0,1},
          x tick label style={opacity=0.4},
          xmin=-1.25,
          xmax=1.25,
          domain=-1:1,
          xlabel={$t$},
          xlabel style={opacity=0.4},
          x axis line style={opacity=0.6},
          ytick={-1,0,1},
          y tick label style={opacity=0.4},
          ymin=-1.25,
          ymax=1.25,
          y domain=-1:1,
          ylabel={$y$},
          ylabel style={opacity=0.4},
          y axis line style={opacity=0.6},
          %ztick={-1,1},
          zticklabels={\empty},
          z tick label style={opacity=0.4},
          zmin=-0.75,
          zmax=0.75,
          zlabel={$z$},
          zlabel style={opacity=0.4},
          z axis line style={opacity=0.6},
        ]
        \coordinate (origin) at (axis cs: 0,0,0); % suggestion of John Kormylo
          \addplot3[surf,color=mc1,opacity=0.2,samples=25,samples y=25,faceted color=mc1!40]{f(x,y)};
          \draw[variable=\t,domain=-1:1,samples=2,color=mc2,thick,dashed] plot (axis cs:0,\t,-\t);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,\t^2,\t);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,-\t^2,-\t);
        \end{axis}
      \end{tikzpicture}
    }
    \qquad
    \subfloat[The parses of the domain of $f$.]{\label{sespex2fig1b}
      \begin{tikzpicture}[baseline=(origin)]% <---- added
        \begin{axis}[
          width=100mm,
          height=60mm,
          view={0}{90},
          axis lines=center,
          xtick={-1,0,1},
          x tick label style={opacity=0.4},
          xmin=-1.25,
          xmax=1.25,
          domain=-1:1,
          xlabel={$t$},
          xlabel style={opacity=0.4},
          x axis line style={opacity=0.6},
          ytick={-1,0,1},
          y tick label style={opacity=0.4},
          ymin=-1.25,
          ymax=1.25,
          y domain=-1:1,
          ylabel={$y$},
          ylabel style={opacity=0.4},
          y axis line style={opacity=0.6},
          ztick={-1,1},
          zticklabels={\empty},
          z tick label style={opacity=0.4},
          zmin=-1.25,
          zmax=1.25,
          zlabel={$z$},
          zlabel style={opacity=0.4},
          z axis line style={opacity=0.6},
        ]
        \coordinate (origin) at (axis cs: 0,0,0); % suggestion of John Kormylo
          \addplot3[surf,color=mc1,opacity=0.2,samples=25,samples y=25,faceted color=mc1!40]{0};
          \draw[variable=\t,domain=-1:1,samples=2,color=mc2,very thick,dashed] plot (axis cs:0,\t,0);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,\t^2,0);
          \draw[variable=\t,domain=0:1,samples=11,color=mc2,thick,dashed] plot (axis cs:\t,-\t^2,0);
        \end{axis}
      \end{tikzpicture}
    }
    \caption{Graphics of $f:\R^{2}\to\R$.}
    \label{sespex2fig1}
\end{figure}
\end{document}

the obtained result is the same. possible advantage is that you have more flexibility where the origin  is in the image (not tested).

