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We use new deep 21 cm HI observations of the moderately inclined galaxy NGC4559 in the HALOGAS survey to investigate the properties of extra-planar gas.We use TiRiFiC to construct simulated data cubes to match the HI observations.We find that a thick disk component of scale height $\sim\,2\,\mathrm{kpc}$,characterized by a negative vertical gradient in its rotation velocity (lag) of$\sim13 \pm 5$ km s$^{-1}$ kpc$^{-1}$ is an adequate fit to extra-planar gasfeatures. The tilted ring models also present evidence for a decrease in themagnitude of the lag outside of $R_{25}$, and a radial inflow of $\sim 10$ kms$^{-1}$. We extracted lagging extra-planar gas through Gaussian velocityprofile fitting. From both the 3D models and and extraction analyses weconclude that $\sim10-20\%$ of the total {\HI} mass is extra-planar. Most ofthe extra-planar gas is spatially coincident with regions of star formation inspiral arms, as traced by H$\alpha$ and GALEX FUV images, so it is likely dueto star formation processes driving a galactic fountain. We also find thesignature of a filament of a kinematically "forbidden" HI, containing $\sim1.4\times 10^{6}$ M$_{\odot}$ of HI, and discuss its potential relationship toa nearby HI hole. We discover a previously undetected dwarf galaxy in HIlocated $\sim 0.4^{\circ}$ ($\sim 58$ kpc) from the center of NGC 4559,containing $\sim 4\times10^{5}$ M$_{\odot}$. This dwarf has counterpart sourcesin SDSS with spectra typical of HII regions, and we conclude it is two mergingblue compact dwarf galaxies.

HALOGAS Observations of NGC 4559: Anomalous and Extra-planar HI and its Relation to Star Formation