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Does there exist an analytic function $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $f$ is bounded on both real and imaginary axes?

According to Liouville's theorem the function must be constant but I am confused that any bounded function $f$ on $\mathbb{C}$ must be like $f(z)=c$ where c is a constant. I am convinced that it must be bounded with respect to one axis but how about other axis. For example, $f(z)=1$ then it is bounded w.r.t real axis but how about imaginary axis? Question may be easy for other but I am unable to visualize it .

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    What do you mean by "bounded with respect to one axis"? $f(z)=1$ is constant, in particular it is bounded - on the whole plane, not with respect to anything... – 2017-01-23
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    I think the OP asks for the existence of a non-constant entire function $f$ with $f\restriction_{\mathbb{R}}$ bounded and $f\restriction_{\mathrm i\mathbb{R}}$ bounded. – 2017-01-23
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    I mean it takes only one value y=1 at y-axis but all values on x-axis by looking at the graph of the function. – 2017-01-23

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