How to express time complexity when the exponential "e" comes into play?
It depends on whether you're talking about $O$ or $\Theta$. The notation $O$ indicates that the complexity is "this much or smaller", and the notation $\Theta$ indicates that the complexity is "this much, no more or less".

The complexity of $0.15 e^{0.05 n}$ is $\Theta(e^{0.05 n})$. The complexity is not $\Theta(e^n)$, because $e^n$ grows faster than $e^{0.05 n}$ does (the ratio between the two increases without bound).

It is, in fact, true that $0.15 e^{0.05 n}$ is an $O(e^n)$ function, but this isn't a good description of the complexity, in much the same way that "more than a thousand people live in China" is not a good description of the population of China.

What this comes down to is that you should state the complexity as $\Theta(e^{0.05 n})$ or as $O(e^{0.05 n})$.